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Calculus Cameron Geisler

Section 2.1 Increasing and Decreasing Functions

Derivatives provide a lot of information about the shape of a function’s graph. First, we will analyze whether functions are increasing or decreasing.
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Subsection 2.1.1 Increasing and Decreasing Functions

Recall what it means for a function to be increasing or decreasing.
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Definition 2.1.1. Increasing and decreasing functions.

  • A function \(f\) is increasing if when \(x\) increases, \(f(x)\) increases.
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  • A function \(f\) is decreasing if when \(x\) increases, \(f(x)\) decreases.
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In other words,
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\begin{gather*} \boxed{\begin{array}{c} \text{increasing} \ \iff x \uparrow y \uparrow \\ \text{decreasing} \ \iff x \uparrow y \downarrow \end{array}} \end{gather*}

Remark 2.1.2.

Increasing and decreasing come from the perspective of reading from left to right (or, with increasing \(x\)).
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Subsection 2.1.2 Increasing/Decreasing Test (Positive Derivative Implies Increasing)

Increasing and decreasing directly relate to derivatives. Recall that the value of the derivative \(f'\) represents the slope of the tangent line of \(f\text{.}\) This means that,
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  • If \(f'(x) \gt 0\text{,}\) then the tangent line has positive slope, and its graph is sloping up to the right, and so is increasing.
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  • Similarly, if \(f'(x) \lt 0\text{,}\) then the tangent line has negative slope, and so \(f\) is decreasing.
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In short,
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\begin{gather*} \boxed{\begin{array}{c} f' \gt 0 \ \iff \text{increasing} \\ f' \lt 0 \ \iff \text{decreasing} \end{array}} \end{gather*}
Graphically, these statements are intuitively true. However, a proof requires the mean value theorem, which we will cover later on.
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Subsection 2.1.3 Finding Intervals of Increase and Decrease

Exercise Group 2.1.1. Basic Examples.

(a)
\(f(x)=4-x^2\)
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Hint.
\(f'(x)=-2x\)
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Answer.
increasing: \((-\infty,0)\text{,}\) decreasing: \((0,\infty)\)
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(b)
\(f(x)=x^3-3x\)
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Hint.
\(f'(x)=3(x-1)(x+1)\)
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Answer.
increasing: \((-\infty,-1),(1,\infty)\text{,}\) decreasing: \((-1,1)\)
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(c)
\(f(x)=2x^3+3x^2+1\)
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Hint.
\(f'(x)=6x(x+1)\)
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Answer.
increasing: \((-\infty,-1),(0,\infty)\text{,}\) decreasing: \((-1,0)\)
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(d)
\(f(x)=x^2-16\)
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Hint.
\(f'(x)=2x\)
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Answer.
increasing: \((0,\infty)\text{,}\) decreasing: \((-\infty,0)\)
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(e)
\(f(x)=x^3+4x\)
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Hint.
\(f'(x)=3x^2+4\)
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Answer.
increasing: \((-\infty,\infty)\text{,}\) decreasing: none
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(f)
\(f(x)=(x-1)^2\)
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Hint.
\(f'(x)=2x-2\)
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Answer.
increasing: \((1,\infty)\text{,}\) decreasing: \((-\infty,1)\)
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(g)
\(f(x)=\frac{x^3}{3}-\frac{5x^2}{2}+4x\)
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Hint.
\(f'(x)=(x-1)(x-4)\)
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Answer.
increasing: \((-\infty,1),(4,\infty)\text{,}\) decreasing: \((1,4)\)
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(h)
\(f(x)=-\frac{x^3}{3}+\frac{x^2}{2}+2x\)
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Hint.
\(f'(x)=-(x-2)(x+1)\)
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Answer.
increasing: \((-1,2)\text{,}\) decreasing: \((-\infty,-1),(2,\infty)\)
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(i)
\(f(x)=12+x-x^2\)
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Hint.
\(f'(x)=1-2x\)
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Answer.
increasing: \((-\infty,\frac{1}{2})\text{,}\) decreasing: \((\frac{1}{2},\infty)\)
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Exercise Group 2.1.2. More Polynomial Examples.

(a)
\(f(x)=x^4-4x^3+4x^2\)
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Hint.
\(f'(x)=4x(x-1)(x-2)\)
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Answer.
increasing: \((0,1),(2,\infty)\text{,}\) decreasing: \((-\infty,0),(1,2)\)
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(b)
\(f(x)=-\frac{x^4}{4}+x^3-x^2\)
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Hint.
\(f'(x)=-x(x-1)(x-2)\)
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Answer.
increasing: \((-\infty,0),(1,2)\text{,}\) decreasing: \((0,1),(2,\infty)\)
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(c)
\(f(x)=3x^4+4x^3-12x^2+2\)
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Hint.
\(f'(x)=12x(x-1)(x+2)\)
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Answer.
increasing: \((-2,0),(1,\infty)\text{,}\) decreasing: \((-\infty,-2),(0,1)\)
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(d)
\(f(x)=2x^5-\frac{15x^4}{4}+\frac{5x^3}{3}\)
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Hint.
\(f'(x)=5x^2(2x-1)(x-1)\)
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Answer.
increasing: \((-\infty,\frac{1}{2}),(1,\infty)\text{,}\) decreasing: \((\frac{1}{2},1)\)
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(e)
\(f(x)=-12x^5+75x^4-80x^3\)
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Hint.
\(f'(x)=-60x^2(x-1)(x-4)\)
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Answer.
increasing: \((1,4)\text{,}\) decreasing: \((-\infty,1),(4,\infty)\)
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(f)
\(f(x)=-2x^4+x^2+10\)
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Hint.
\(f'(x)=-2x(2x-1)(2x+1)\)
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Answer.
increasing: \((-\infty,-\frac{1}{2}),(0,\frac{1}{2})\text{,}\) decreasing: \((-\frac{1}{2},0),(\frac{1}{2},\infty)\)
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(g)
\(f(x)=\frac{x^4}{4}-\frac{8x^3}{3}+\frac{15x^2}{2}+8\)
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Hint.
\(f'(x)=x(x-3)(x-5)\)
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Answer.
increasing: \((0,3),(5,\infty)\text{,}\) decreasing: \((-\infty,0),(3,5)\)
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Exercise Group 2.1.3. Trigonometry Examples.

(a)
\(f(x)=-2\cos x-x\) on \([0,2\pi]\)
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Hint.
\(f'(x)=2\sin x-1\)
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Answer.
increasing: \((\frac{\pi}{6},\frac{5\pi}{6})\text{,}\) decreasing: \((0,\frac{\pi}{6}),(\frac{5\pi}{6},2\pi)\)
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(b)
\(f(x)=\sqrt{2}\sin x-x\) on \([0,2\pi]\)
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Hint.
\(f'(x)=\sqrt{2}\cos x-1\)
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Answer.
increasing: \((0,\frac{\pi}{4}),(\frac{7\pi}{4},2\pi)\text{,}\) decreasing: \((\frac{\pi}{4},\frac{7\pi}{4})\)
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(c)
\(f(x)=\cos^2 x\) on \([-\pi,\pi]\)
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Hint.
\(f'(x)=-2\sin x\cos x=-\sin 2x\)
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Answer.
increasing: \((-\frac{\pi}{2},0),(\frac{\pi}{2},\pi)\text{,}\) decreasing: \((-\pi,-\frac{\pi}{2}),(0,\frac{\pi}{2})\)
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(d)
\(f(x)=3\cos 3x\) on \([-\pi,\pi]\)
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Hint.
\(f'(x)=-9\sin 3x\)
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Answer.
increasing: \((-\pi,-\frac{2\pi}{3}),(-\frac{\pi}{3},0),(\frac{\pi}{3},\frac{2\pi}{3})\text{,}\) decreasing: \((-\frac{2\pi}{3},-\frac{\pi}{3}),(0,\frac{\pi}{3}),(\frac{2\pi}{3},\pi)\)
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Exercise Group 2.1.4. More Examples.

(a)
\(f(x)=(x^2-4)^{1/2}\)
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Hint.
\(f'(x)=\frac{x}{\sqrt{x^2-4}}\)
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Answer.
increasing: \((2,\infty)\text{,}\) decreasing: \((-\infty,-2)\)
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(b)
\(f(x)=xe^{-x}\)
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Hint.
\(f'(x)=e^{-x}(1-x)\)
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Answer.
increasing: \((-\infty,1)\text{,}\) decreasing: \((1,\infty)\)
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(c)
\(f(x)=3xe^x\)
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Hint.
\(f'(x)=3e^x(x+1)\)
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Answer.
increasing: \((-1,\infty)\text{,}\) decreasing: \((-\infty,-1)\)
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(d)
\(f(x)=x^2-2\ln x\)
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Hint.
\(f'(x)=2x-\frac{2}{x}=\frac{2(x-1)(x+1)}{x}\)
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Answer.
increasing: \((1,\infty)\text{,}\) decreasing: \((0,1)\)
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(e)
\(f(x)=\sqrt{x}\cdot\ln x\)
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Hint.
\(f'(x)=\frac{\ln x+2}{2\sqrt{x}}\)
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Answer.
increasing: \((\frac{1}{e^2},\infty)\text{,}\) decreasing: \((0,\frac{1}{e^2})\)
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(f)
\(f(x)=\frac{e^x}{e^{2x}+1}\)
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Hint.
\(f'(x)=\frac{e^x(1-e^{2x})}{(e^{2x}+1)^2}\)
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Answer.
increasing: \((-\infty,0)\text{,}\) decreasing: \((0,\infty)\)
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(g)
\(f(x)=x\ln x-2x+3\) on \((0,\infty)\)
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Hint.
\(f'(x)=\ln x-1\)
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Answer.
increasing: \((e,\infty)\text{,}\) decreasing: \((0,e)\)
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(h)
\(f(x)=xe^{-x^2/2}\)
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Hint.
\(f'(x)=e^{-x^2/2}(1-x^2)\)
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Answer.
increasing: \((-1,1)\text{,}\) decreasing: \((-\infty,-1),(1,\infty)\)
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(i)
\(f(x)=\frac{x}{x^2+1}\)
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Hint.
\(f'(x)=\frac{1-x^2}{(x^2+1)^2}\)
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Answer.
increasing: \((-1,1)\text{,}\) decreasing: \((-\infty,-1),(1,\infty)\)
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(j)
\(f(x)=\sqrt{x^2+4}\)
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Hint.
\(f'(x)=\frac{x}{\sqrt{x^2+4}}\)
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Answer.
increasing: \((0,\infty)\text{,}\) decreasing: \((-\infty,0)\)
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(k)
\(f(x)=\frac{5x}{x^2+1}\)
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Hint.
\(f'(x)=\frac{-5(x^2-1)}{(x^2+1)^2}\)
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Answer.
increasing: \((-1,1)\text{,}\) decreasing: \((-\infty,-1),(1,\infty)\)
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(l)
\(f(x)=\frac{2x}{x^2+9}\)
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Hint.
\(f'(x)=\frac{2(9-x^2)}{(x^2+9)^2}=-\frac{2(x-3)(x+3)}{(x^2+9)^2}\)
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Answer.
increasing: \((-3,3)\text{,}\) decreasing: \((-\infty,-3),(3,\infty)\)
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(m)
\(f(x)=x+\frac{1}{x}\)
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Hint.
\(f'(x)=1-\frac{1}{x^2}=\frac{x^2-1}{x^2}\)
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Answer.
increasing: \((-\infty,-1),(1,\infty)\text{,}\) decreasing: \((-1,0),(0,1)\)
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(n)
\(f(x)=\frac{x-1}{x^2+3}\)
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Hint.
\(f'(x)=-\frac{(x-3)(x+1)}{(x^2+3)^2}\)
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Answer.
increasing: \((-1,3)\text{,}\) decreasing: \((-\infty,-1),(3,\infty)\)
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Exercise Group 2.1.5. Advanced Examples.

(a)
\(f(x)=(2x-1)^2(x^2-9)\)
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Hint.
\(f'(x)=2(2x-1)(4x-9)(x+2)\)
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Answer.
increasing: \((-2,\frac{1}{2}),(\frac{9}{4},\infty)\text{,}\) decreasing: \((-\infty,-2),(\frac{1}{2},\frac{9}{4})\)
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(b)
\(f(x)=\tan^{-1}\brac{\frac{x}{x^2+2}}\)
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Hint.
\(f'(x)=\frac{2-x^2}{x^4+5x^2+4}\)
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Answer.
increasing: \((-\sqrt{2},\sqrt{2})\text{,}\) decreasing: \((-\infty,-\sqrt{2}),(\sqrt{2},\infty)\)
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(c)
\(f(x)=\sqrt{9-x^2}+\sin^{-1}\brac{\frac{x}{3}}\)
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Hint.
\(f'(x)=\frac{1-x}{\sqrt{9-x^2}}\)
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Answer.
increasing: \((-3,1)\text{,}\) decreasing: \((1,3)\)
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