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Section 2.3 Sketching Graphs of Functions
Subsection 2.3.1 Examples
Exercise Group 2.3.1 . Polynomials.
Sketch the graph of each function.
(a)
Hint .
\(f'(x)=2x-4\text{,}\) \(f''(x)=2\text{.}\)
Answer .
increasing:
\((2,\infty)\text{,}\) decreasing:
\((-\infty,2)\text{,}\) local maxima: none, local minima:
\((2,-1)\text{,}\) concave up:
\((-\infty,\infty)\text{,}\) concave down: none, inflection point(s): none.
(b)
Hint .
\(f'(x)=3(x-1)(x+1)\text{,}\) \(f''(x)=6x\text{.}\)
Answer .
increasing:
\((-\infty,-1),(1,\infty)\text{,}\) decreasing:
\((-1,1)\text{,}\) local maxima:
\((-1,5)\text{,}\) local minima:
\((1,1)\text{,}\) concave up:
\((0,\infty)\text{,}\) concave down:
\((-\infty,0)\text{,}\) inflection point(s):
\((0,3)\text{.}\)
(c)
Hint .
\(f'(x)=6(x-1)(x-2)\text{,}\) \(f''(x)=12x-18\text{.}\)
Answer .
increasing:
\((-\infty,1),(2,\infty)\text{,}\) decreasing:
\((1,2)\text{,}\) local maxima:
\((1,2)\text{,}\) local minima:
\((2,1)\text{,}\) concave down:
\((-\infty,\frac{3}{2})\text{,}\) concave up:
\((\frac{3}{2},\infty)\text{,}\) inflection point(s):
\(\brac{\frac{3}{2},\frac{3}{2}}\text{.}\)
(d)
Hint .
\(f'(x)=4x(x-1)(x+1)\text{,}\) \(f''(x)=4(3x^2-1)\text{.}\)
Answer .
increasing:
\((-1,0),(1,\infty)\text{,}\) decreasing:
\((-\infty,-1),(0,1)\text{,}\) local maxima:
\((0,0)\text{,}\) local minima:
\((-1,-1),(1,-1)\text{,}\) concave up:
\((-\infty,-\frac{\sqrt3}{3}),(\frac{\sqrt3}{3},\infty)\text{,}\) concave down
\((-\frac{\sqrt3}{3},\frac{\sqrt3}{3})\text{,}\) inflection point(s):
\((\pm\frac{\sqrt3}{3},-\frac{5}{9})\text{.}\)
(e)
Hint .
\(f'(x)=4(x-4)^2(x-1)\text{,}\) \(f''(x)=12(x-4)(x-2)\text{.}\)
Answer .
\(x\) -intercepts
\(x=0,4\text{,}\) \(y\) -intercept:
\(y=0\text{,}\) increasing:
\((1,\infty)\text{,}\) decreasing:
\((-\infty,1)\text{,}\) local minima:
\((1,-27)\text{,}\) local maxima: none, concave up:
\((-\infty,2),(4,\infty)\text{,}\) concave down:
\((2,4)\text{,}\) inflection point(s):
\((2,-16),(4,0)\text{.}\)
Exercise Group 2.3.2 . Rational Functions (Vertical Asymptotes).
Sketch the graph of each function.
(a)
Hint .
\(f'(x)=\frac{x^2+9}{(9-x^2)^2}\text{,}\) \(f''(x)=-\frac{2x(x^2+27)}{(9-x^2)^3}\text{.}\)
Answer .
domain:
\((-\infty,-3),(-3,3),(3,\infty)\text{,}\) \(x\) -intercepts
\(x=0\text{,}\) \(y\) -intercept:
\(y=0\text{,}\) increasing:
\((-\infty,-3),(-3,3),(3,\infty)\text{,}\) decreasing: none, local minima: none, local maxima: none, concave up:
\((-\infty,-3),(0,3)\text{,}\) concave down:
\((-3,0),(3,\infty)\text{.}\)
(b)
\(f(x)=\frac{x}{x^2-4x+4}\)
Hint .
\(f(x)=\frac{x}{(x-2)^2}\text{,}\) \(f'(x)=\frac{-(x+2)}{(x-2)^3}\text{,}\) \(f''(x)=\frac{2(x+4)}{(x-2)^4}\text{.}\)
Answer .
domain:
\((-\infty,2)\cup(2,\infty)\text{,}\) \(x\) -intercept:
\(x=0\text{,}\) \(y\) -intercept:
\(y=0\text{,}\) vertical asymptote
\(x=2\text{,}\) horizontal asymptote
\(y=0\text{,}\) increasing
\((-2,2)\text{,}\) decreasing
\((-\infty,-2)\cup(2,\infty)\text{,}\) local minima:
\((-2,-\frac{1}{8})\text{,}\) local maxima: none, concave down:
\((-\infty,-4)\text{,}\) concave up:
\((-4,2)\cup(2,\infty)\text{,}\) inflection point:
\((-4,-\frac{1}{9})\text{.}\)
(c)
Hint .
\(f'(x)=\frac{1-x^2}{(x^2+1)^2}\text{,}\) \(f''(x)=\frac{2x(x^2-3)}{(x^2+1)^3}\text{.}\)
Answer .
domain:
\((-\infty,\infty)\text{,}\) \(x\) -intercept
\(x=0\text{,}\) \(y\) -intercept:
\(y=0\text{,}\) increasing:
\((-1,1)\text{,}\) decreasing:
\((-\infty,-1),(1,\infty)\text{,}\) local max:
\((1,\frac{1}{2})\text{,}\) local min:
\((-1,-\frac{1}{2})\text{,}\) concave up:
\((-\sqrt{3},0),(\sqrt{3},\infty)\text{,}\) concave down:
\((-\infty,-\sqrt{3}),(0,\sqrt{3})\text{,}\) inflection points:
\(\brac{-\sqrt{3},-\frac{\sqrt{3}}{4}},(0,0),\brac{\sqrt{3},\frac{\sqrt{3}}{4}}\text{.}\)
(d)
\(f(x)=\frac{x^2-49}{x^2+5x-14}\)
Hint .
\(f'(x)=\frac{5}{(x-2)^2}\text{,}\) \(f''(x)=-\frac{10}{(x-2)^3}\text{.}\)
Answer .
domain
\((-\infty,-7),(-7,2),(2,\infty)\text{,}\) \(x\) -intercept
\(x=7\text{,}\) \(y\) -intercept:
\(y=\frac{7}{2}\text{,}\) vertical asymptote:
\(x=2\text{,}\) hole:
\((-7,\frac{14}{9})\text{,}\) horizontal asymptote:
\(y=1\text{,}\) increasing:
\((-\infty,-7),(-7,2),(2,\infty)\text{,}\) decreasing: none, local minima: none, local maxima: none, concave up:
\((-\infty,-7),(-7,2)\text{,}\) concave down:
\((2,\infty)\text{.}\)
Exercise Group 2.3.3 . Trigonometric Functions.
Sketch the graph of each function.
(a)
\(f(x)=\sin x+\cos x\text{,}\) \(0\leq x\leq 2\pi\)
Hint .
\(f'(x)=\cos x-\sin x\text{,}\) \(f''(x)=-(\sin x+\cos x)\text{.}\)
Answer .
\(x\) -intercepts
\(x=\frac{3\pi}{4},\frac{7\pi}{4}\text{,}\) \(y\) -intercept:
\(y=1\text{,}\) increasing:
\((0,\frac{\pi}{4}),(\frac{5\pi}{4},2\pi)\text{,}\) decreasing:
\((\frac{\pi}{4},\frac{5\pi}{4})\text{,}\) local max:
\(\brac{\frac{\pi}{4},\sqrt{2}}\text{,}\) local min:
\(\brac{\frac{5\pi}{4},-\sqrt{2}}\text{,}\) concave down:
\((0,\frac{3\pi}{4}),(\frac{7\pi}{4},2\pi)\text{,}\) concave up:
\((\frac{3\pi}{4},\frac{7\pi}{4})\text{,}\) inflection points:
\((\frac{3\pi}{4},0),(\frac{7\pi}{4},0)\text{.}\)
(b)
\(f(x)=2\cos x+\cos^2 x\text{,}\) \(0\leq x\leq 2\pi\)
Hint .
\(f'(x)=-2\sin x(1+\cos x)\text{,}\) \(f''(x)=-2(\cos x+\cos 2x)\text{.}\)
Answer .
\(x\) -intercepts
\(x=\frac{\pi}{2},\frac{3\pi}{2}\text{,}\) \(y\) -intercept 3, decreasing:
\((0,\pi)\text{,}\) increasing:
\((\pi,2\pi)\text{,}\) local min:
\((\pi,-1)\text{,}\) local maxima: none, concave down:
\((0,\frac{\pi}{3}),(\frac{5\pi}{3},2\pi)\text{,}\) concave up:
\((\frac{\pi}{3},\frac{5\pi}{3})\text{,}\) inflection points:
\(\brac{\frac{\pi}{3},\frac{5}{4}},\brac{\frac{5\pi}{3},\frac{5}{4}}\text{.}\)
(c)
\(f(x)=x+\sin x\text{,}\) \(0\leq x\leq 2\pi\)
Hint .
\(f'(x)=1+\cos x\text{,}\) \(f''(x)=-\sin x\text{.}\)
Answer .
domain
\([0,2\pi]\text{,}\) \(x\) -intercepts
\(x=0\text{,}\) \(y\) -intercept 0, increasing
\((0,\pi),(\pi,2\pi)\text{,}\) decreasing none, local minima none, local maxima none, concave down
\((0,\pi)\text{,}\) concave up
\((\pi,2\pi)\text{,}\) inflection point
\((\pi,\pi)\text{.}\)
(d)
\(f(x)=\sin x\cos x\text{,}\) \(0\leq x\leq \pi\)
Hint .
\(f'(x)=\cos^2{x}-\sin^2{x}\text{,}\) \(f''(x)=-4\sin{x}\cos{x}\text{.}\)
Answer .
domain
\([0,\pi]\text{,}\) \(x\) -intercepts
\(x=0,\frac{\pi}{2},\pi\text{,}\) \(y\) -intercept 0, increasing
\((0,\frac{\pi}{4}),(\frac{3\pi}{4},\pi)\text{,}\) decreasing
\((\frac{\pi}{4},\frac{3\pi}{4})\text{,}\) local maxima
\((\frac{\pi}{4},\frac{1}{2})\text{,}\) local minima
\((\frac{3\pi}{4},-\frac{1}{2})\text{,}\) concave down
\((0,\frac{\pi}{2})\text{,}\) concave up
\((\frac{\pi}{2},\pi)\text{,}\) inflection point
\((\frac{\pi}{2},0)\text{.}\)
(e)
\(f(x)=x-\sin x\text{,}\) \(0\leq x\leq 2\pi\)
Hint .
\(f'(x)=1-\cos x\text{,}\) \(f''(x)=\sin x\text{.}\)
Answer .
increasing
\((0,2\pi)\text{,}\) decreasing none, local minima none, local maxima none, concave up
\((0,\pi)\text{,}\) concave down
\((\pi,2\pi)\text{,}\) inflection point
\((\pi,\pi)\text{.}\)
(f)
\(f(x)=\sqrt3 x-2\cos x\text{,}\) \(0\leq x\leq 2\pi\)
Hint .
\(f'(x)=\sqrt3+2\sin x\text{,}\) \(f''(x)=2\cos x\text{.}\)
Answer .
domain
\([0,2\pi]\text{,}\) \(y\) -intercept
\(-2\text{,}\) increasing
\((0,\frac{4\pi}{3}),(\frac{5\pi}{3},2\pi)\text{,}\) decreasing
\((\frac{4\pi}{3},\frac{5\pi}{3})\text{,}\) local maxima
\((\frac{4\pi}{3},\frac{4\pi\sqrt3}{3}+1)\text{,}\) local minima
\((\frac{5\pi}{3},\frac{5\pi\sqrt3}{3}-1)\text{,}\) concave up
\((0,\frac{\pi}{2}),(\frac{3\pi}{2},2\pi)\text{,}\) concave down
\((\frac{\pi}{2},\frac{3\pi}{2})\text{,}\) inflection points
\((\frac{\pi}{2},\frac{\pi\sqrt3}{2})\) and
\((\frac{3\pi}{2},\frac{3\pi\sqrt3}{2})\text{.}\)
(g)
\(f(x)=\cos^2 x-2\sin x\text{,}\) \(0\leq x\leq 2\pi\)
Hint .
\(f'(x)=-2\cos x(\sin x+1)\text{,}\) \(f''(x)=2(\sin x-\cos 2x)\text{.}\)
Answer .
\(y\) -intercept 1, increasing:
\((\frac{\pi}{2},\frac{3\pi}{2})\text{,}\) decreasing:
\((0,\frac{\pi}{2}),(\frac{3\pi}{2},2\pi)\text{,}\) local min:
\(\brac{\frac{\pi}{2},-2}\text{,}\) local max:
\((\frac{3\pi}{2},2)\text{,}\) concave down:
\((0,\frac{\pi}{6}),(\frac{5\pi}{6},2\pi)\text{,}\) concave up:
\((\frac{\pi}{6},\frac{5\pi}{6})\text{,}\) inflection points:
\(\brac{\frac{\pi}{6},\frac{5}{4}},\brac{\frac{5\pi}{6},-\frac{3}{4}}\text{.}\)
Exercise Group 2.3.4 . Vertical Tangents and Cusps.
Sketch the graph of each function.
(a)
Hint .
\(f'(x) = 2-\frac{2}{\sqrt[3]{x}}\text{,}\) \(f''(x) = \frac{2}{3x^{4/3}}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercepts
\(x = 0,\frac{27}{8}\text{,}\) \(y\) -intercept 0, increasing:
\((-\infty,0),(1,\infty)\text{,}\) decreasing:
\((0,1)\text{,}\) local minima:
\((1,-1)\text{,}\) local maxima:
\((0,0)\text{,}\) concave up:
\((-\infty,0),(0,\infty)\text{,}\) concave down: none.
(b)
Hint .
\(f'(x)=\frac{2}{3(x-4)^{1/3}}\text{,}\) \(f''(x)=\frac{-2}{9(x-4)^{4/3}}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercept
\(x=4\text{,}\) \(y\) -intercept
\(4^{2/3}\text{,}\) decreasing
\((-\infty,4)\text{,}\) increasing
\((4,\infty)\text{,}\) local minimum
\((4,0)\text{,}\) concave down
\((-\infty,4),(4,\infty)\text{,}\) concave up none, inflection points none.
(c)
Hint .
\(f'(x) = \frac{2}{x^{3/5}}-2\text{,}\) \(f''(x) = -\frac{6}{5x^{8/5}}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercepts
\(x = 0,\left(\frac{5}{2}\right)^{5/3}\text{,}\) \(y\) -intercept 0, increasing:
\((0,1)\text{,}\) decreasing:
\((-\infty,0),(1,\infty)\text{,}\) local minima:
\((0,0)\text{,}\) local maxima:
\((1,3)\text{,}\) concave up: none, concave down:
\((-\infty,0),(0,\infty)\text{.}\)
(d)
\(f(x)=x^{1/3}(x+3)^{2/3}\)
Hint .
\(f'(x)=\frac{x+1}{x^{2/3}(x+3)^{1/3}}\text{,}\) \(f''(x)=\frac{-2}{x^{5/3}(x+3)^{4/3}}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercepts
\(x=-3,0\text{,}\) \(y\) -intercept 0, increasing
\((-\infty,-3),(-1,0),(0,\infty)\text{,}\) decreasing
\((-3,-1)\text{,}\) local maximum
\((-3,0)\text{,}\) local minimum
\((-1,-2^{2/3})\text{,}\) concave up
\((-\infty,0)\text{,}\) concave down
\((0,\infty)\text{,}\) inflection point
\((0,0)\text{.}\)
Exercise Group 2.3.5 . Slant Asymptotes.
Sketch the graph of each function.
(a)
Hint .
\(f'(x)=\frac{x(x+2)}{(x+1)^2}\text{,}\) \(f''(x)=\frac{2}{(x+1)^3}\text{.}\)
Answer .
domain:
\((-\infty,-1)\cup(-1,\infty)\text{,}\) \(x\) -intercept:
\(x=0\text{,}\) \(y\) -intercept:
\(y=0\text{,}\) vertical asymptote:
\(x=-1\text{,}\) slant asymptote:
\(y=x-1\text{,}\) increasing
\((-\infty,-2),(0,\infty)\text{,}\) decreasing
\((-2,-1),(-1,0)\text{,}\) local maxima
\((-2,-4)\text{,}\) local minima
\((0,0)\text{,}\) concave down
\((-\infty,-1)\text{,}\) concave up
\((-1,\infty)\text{.}\)
(b)
\(f(x)=\frac{x^2-x+1}{x-1}\)
Hint .
\(f'(x)=\frac{x(x-2)}{(x-1)^2}\text{,}\) \(f''(x)=\frac{2}{(x-1)^3}\text{.}\)
Answer .
domain
\((-\infty,1)\cup(1,\infty)\text{,}\) \(x\) -intercepts none,
\(y\) -intercept
\(-1\text{,}\) vertical asymptote
\(x=1\text{,}\) slant asymptote
\(y=x\text{,}\) increasing
\((-\infty,0)\cup(2,\infty)\text{,}\) decreasing
\((0,1)\cup(1,2)\text{,}\) local maxima
\((0,-1)\text{,}\) local minima
\((2,3)\text{,}\) concave down
\((-\infty,1)\text{,}\) concave up
\((1,\infty)\text{.}\)
(c)
\(f(x)=\frac{x^2-3x+2}{x-3}\)
Hint .
\(f(x)=x+\frac{2}{x-3}\text{,}\) \(f'(x)=1-\frac{2}{(x-3)^2}\text{,}\) \(f''(x)=\frac{4}{(x-3)^3}\text{.}\)
Answer .
domain
\((-\infty,3)\cup(3,\infty)\text{,}\) \(x\) -intercepts
\(x=1,2\text{,}\) \(y\) -intercept
\(-\frac{2}{3}\text{,}\) vertical asymptote
\(x=3\text{,}\) slant asymptote
\(y=x\text{,}\) increasing
\((-\infty,3-\sqrt{2})\cup(3+\sqrt{2},\infty)\text{,}\) decreasing
\((3-\sqrt{2},3)\cup(3,3+\sqrt{2})\text{,}\) local maximum
\((3-\sqrt{2},3-2\sqrt{2})\text{,}\) local minimum
\((3+\sqrt{2},3+2\sqrt{2})\text{,}\) concave down
\((-\infty,3)\text{,}\) concave up
\((3,\infty)\text{,}\) inflection points none.
(d)
\(f(x)=\frac{x^3-3x^2+3x-1}{x^2+x-2}\)
Hint .
\(f'(x)=\frac{(x-1)(x+5)}{(x+2)^2}\text{,}\) \(f''(x)=\frac{18}{(x+2)^3}\text{.}\)
Answer .
domain
\((-\infty,-2)\cup(-2,1)\cup(1,\infty)\text{,}\) \(x\) -intercepts none,
\(y\) -intercept
\(\frac{1}{2}\text{,}\) vertical asymptote
\(x=-2\text{,}\) hole
\((1,0)\text{,}\) slant asymptote
\(y=x-4\text{,}\) increasing
\((-\infty,-5)\cup(1,\infty)\text{,}\) decreasing
\((-5,-2)\cup(-2,1)\text{,}\) local maxima
\((-5,-12)\text{,}\) local minima none, concave down
\((-\infty,-2)\text{,}\) concave up
\((-2,1)\cup(1,\infty)\text{.}\)
Exercise Group 2.3.6 . Polynomial and Rational Functions.
Sketch the graph of each function.
(a)
Hint .
\(f'(x)=12(x-3)(x-1)\text{,}\) \(f''(x)=24x-48\text{.}\)
Answer .
increasing:
\((-\infty,1),(3,\infty)\text{,}\) decreasing:
\((1,3)\text{,}\) local maxima:
\((1,16)\text{,}\) local minima:
\((3,0)\text{,}\) concave up:
\((2,\infty)\text{,}\) concave down:
\((-\infty,2)\text{,}\) inflection point(s):
\((2,8)\text{.}\)
(b)
\(f(x)=\frac{4x}{x^2+4}\)
Hint .
\(f'(x)=-\frac{4(x-2)(x+2)}{(x^2+4)^2}\text{,}\) \(f''(x)=\frac{8x(x^2-12)}{(x^2+4)^3}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercept
\((0,0)\text{,}\) \(y\) -intercept 0, horizontal asymptote
\(y=0\text{,}\) increasing
\((-2,2)\text{,}\) decreasing
\((-\infty,-2),(2,\infty)\text{,}\) local minima
\((-2,-1)\text{,}\) local maxima
\((2,1)\text{,}\) concave up
\((-2\sqrt3,0),(2\sqrt3,\infty)\text{,}\) concave down
\((-\infty,-2\sqrt3),(0,2\sqrt3)\text{,}\) inflection points
\((-2\sqrt3,-\frac{\sqrt3}{2})\text{,}\) \((0,0)\text{,}\) \((2\sqrt3,\frac{\sqrt3}{2})\text{.}\)
(c)
Hint .
\(f'(x)=\frac{-x^2-1}{(x^2-1)^2}\text{,}\) \(f''(x)=\frac{2x(x^2+3)}{(x-1)^3(x+1)^3}\text{.}\)
Answer .
domain
\((-\infty,-1)\cup(-1,1)\cup(1,\infty)\text{,}\) \(x\) -intercept: 0,
\(y\) -intercept: 0, vertical asymptotes:
\(x=-1,1\text{,}\) horizontal asymptote:
\(y=0\text{,}\) increasing: none, decreasing:
\((-\infty,-1),(-1,1),(1,\infty)\text{,}\) local maxima: none, local minima: none, concave up
\((-1,0),(1,\infty)\text{,}\) concave down
\((-\infty,-1),(0,1)\text{,}\) inflection point
\((0,0)\text{.}\)
(d)
\(f(x)=\frac{x^4+1}{x^2}\)
Hint .
\(f'(x)=\frac{2(x-1)(x+1)(x^2+1)}{x^3}\text{,}\) \(f''(x)=\frac{2(x^4+3)}{x^4}\text{.}\)
Answer .
domain
\((-\infty,0)\cup(0,\infty)\text{,}\) \(x\) -intercepts none,
\(y\) -intercept none, vertical asymptote
\(x=0\text{,}\) increasing
\((-1,0),(1,\infty)\text{,}\) decreasing
\((-\infty,-1),(0,1)\text{,}\) local minima
\((-1,2)\) and
\((1,2)\text{,}\) local maxima none, concave up
\((-\infty,0),(0,\infty)\text{,}\) concave down none.
(e)
\(f(x)=\frac{2x^2}{x^2-1}\)
Hint .
\(f'(x)=\frac{-4x}{(x^2-1)^2}\text{,}\) \(f''(x)=\frac{4(3x^2+1)}{(x^2-1)^3}\text{.}\)
Answer .
domain
\((-\infty,-1)\cup(-1,1)\cup(1,\infty)\text{,}\) \(x\) -intercepts:
\(x=0\text{,}\) \(y\) -intercept:
\(0\text{,}\) increasing:
\((-\infty,-1),(-1,0)\text{,}\) decreasing:
\((0,1),(1,\infty)\text{,}\) local minima: none, local maxima:
\((0,0)\text{,}\) concave up:
\((-\infty,-1),(1,\infty)\text{,}\) concave down:
\((-1,1)\text{,}\) inflection point(s): none.
(f)
\(f(x) = \frac{2x}{5+x^2}\)
Answer .
increasing:
\(\brac{-\sqrt{5},\sqrt{5}}\text{,}\) decreasing:
\((-\infty,-\sqrt{5}),(\sqrt{5},\infty)\text{,}\) local minima:
\(\brac{-\sqrt{5},-\frac{\sqrt{5}}{5}}\text{,}\) local maxima:
\(\brac{\sqrt{5},\frac{\sqrt{5}}{5}}\text{,}\) concave up:
\(\brac{-\sqrt{15},0},(\sqrt{15},\infty)\text{,}\) concave down:
\((-\infty,-\sqrt{15}),(0,\sqrt{15})\text{,}\) inflection point(s):
\(\brac{-\sqrt{15},-\frac{\sqrt{15}}{10}},(0,0),\brac{\sqrt{15},\frac{\sqrt{15}}{10}}\text{.}\)
Exercise Group 2.3.7 . Transcendental Functions.
Sketch the graph of each function.
(a)
Hint .
\(f'(x)=\frac{4-2x^2}{\sqrt{4-x^2}}, f''(x)=\frac{2x^3-12x}{(4-x^2)^{3/2}}\text{.}\)
Answer .
domain
\([-2,2]\text{,}\) \(x\) -intercepts
\(x=-2,0,2\text{,}\) \(y\) -intercept 0, increasing on
\((-\sqrt{2},\sqrt{2})\text{,}\) decreasing:
\((-2,-\sqrt{2}),(\sqrt{2},2)\text{,}\) local minimum:
\((-\sqrt{2},-2)\text{,}\) local maximum:
\((\sqrt{2},2)\text{,}\) concave up:
\((-2,0)\text{,}\) concave down:
\((0,2)\text{,}\) inflection point:
\((0,0)\text{.}\)
(b)
Hint .
\(f'(x)=2e^{2x}-e^{-x}\text{,}\) \(f''(x)=4e^{2x}+e^{-x}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercepts none,
\(y\) -intercept 2, decreasing:
\((-\infty,-\frac{\ln 2}{3})\text{,}\) increasing:
\((-\frac{\ln 2}{3},\infty)\text{,}\) local min:
\(\brac{-\frac{\ln 2}{3},\frac{3}{2^{2/3}}}\text{,}\) local max: none, concave up:
\((-\infty,\infty)\text{,}\) concave down: none, inflection points: none.
(c)
Hint .
\(f'(x)=\frac{2x+1}{2\sqrt{x^2+x-2}}\text{,}\) \(f''(x)=-\frac{9}{4(x^2+x-2)^{3/2}}\text{.}\)
Answer .
domain
\((-\infty,-2]\cup[1,\infty)\text{,}\) \(x\) -intercepts
\(x=-2,1\text{,}\) \(y\) -intercept none, increasing:
\((1,\infty)\text{,}\) decreasing:
\((-\infty,-2)\text{,}\) local minima:
\((-2,0),(1,0)\text{,}\) local maxima: none, concave up: none, concave down:
\((-\infty,-2),(1,\infty)\text{,}\) inflection point(s): none.
(d)
Hint .
\(f'(x)=\frac{2x}{x^2+9}\text{,}\) \(f''(x)=\frac{2(9-x^2)}{(x^2+9)^2}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercepts none,
\(y\) -intercept
\(\ln 9\text{,}\) decreasing:
\((-\infty,0)\text{,}\) increasing:
\((0,\infty)\text{,}\) local min:
\((0,\ln 9)\text{,}\) local max: none, concave up:
\((-3,3)\text{,}\) concave down:
\((-\infty,-3),(3,\infty)\text{,}\) inflection points:
\((-3,\ln 18),(3,\ln 18)\text{.}\)
(e)
\(f(x)=\frac{1}{1+e^{-x}}=\frac{e^x}{1+e^x}\)
Hint .
\(f'(x)=\frac{e^{-x}}{(1+e^{-x})^2}\text{,}\) \(f''(x)=\frac{e^{-x}(e^{-x}-1)}{(1+e^{-x})^3}\text{.}\)
Answer .
domain
\((-\infty,\infty)\text{,}\) \(x\) -intercepts none,
\(y\) -intercept
\(\frac{1}{2}\text{,}\) increasing:
\((-\infty,\infty)\text{,}\) decreasing: none, local minima: none, local maxima: none, concave up:
\((-\infty,0)\text{,}\) concave down:
\((0,\infty)\text{.}\)
(f)
\(f(x)=\frac{\ln{x}}{x^2}\)
Hint .
\(f'(x)=\frac{1-2\ln x}{x^3}\text{,}\) \(f''(x)=\frac{6\ln x-5}{x^4}\text{.}\)
Answer .
domain
\((0,\infty)\text{,}\) \(x\) -intercepts
\(x=1\text{,}\) \(y\) -intercept none, increasing:
\((0,\sqrt{e})\text{,}\) decreasing:
\((\sqrt{e},\infty)\text{,}\) local minima: none, local maxima:
\(\brac{\sqrt{e},\frac{1}{2e}}\text{,}\) concave up:
\((e^{5/6},\infty)\text{,}\) concave down:
\((0,e^{5/6})\text{,}\) inflection point(s):
\(\brac{e^{5/6},\frac{5}{6e^{5/3}}}\text{.}\)
