Local Extrema and the First Derivative Test

increasing: \((-\infty,-\frac{3}{2})\text{,}\) decreasing: \((-\frac{3}{2},\infty)\text{,}\) local maxima: \(\brac{-\frac{3}{2}, \frac{21}{4}}\text{,}\) local minima: none

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(c)

\(f(x)=x^3-6x^2+9x+2\)

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Hint.

\(f'(x)=3x^2-12x+9=3(x-1)(x-3)\)

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Answer.

increasing: \((-\infty,1),(3,\infty)\text{,}\) decreasing: \((1,3)\text{,}\) local maxima: \((1,6)\text{,}\) local minima: \((3,2)\)

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(d)

\(f(x)=(x^2-3)e^x\)

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Hint.

\(f'(x)=e^x(x^2+2x-3)=e^x(x+3)(x-1)\)

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Answer.

increasing: \((-\infty,-3),(1,\infty)\text{,}\) decreasing: \((-3,1)\text{,}\) local maxima: \((-3,\frac{6}{e^3})\text{,}\) local minima: \((1,-2e)\)

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(e)

\(f(x)=4x-x^3\)

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Hint.

\(f'(x)=4-3x^2\)

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Answer.

increasing: \((-\frac{2\sqrt{3}}{3},\frac{2\sqrt{3}}{3})\text{,}\) decreasing: \((-\infty,-\frac{2\sqrt{3}}{3}),(\frac{2\sqrt{3}}{3},\infty)\text{,}\) local maxima: \(\brac{\frac{2\sqrt{3}}{3}, \frac{16\sqrt{3}}{9}}\text{,}\) local minima: \(\brac{-\frac{2\sqrt{3}}{3}, -\frac{16\sqrt{3}}{9}}\)

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(f)

\(f(x)=-3x^3-5x\)

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Hint.

\(f'(x)=-9x^2-5\)

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Answer.

increasing: none, decreasing: \((-\infty,\infty)\text{,}\) local maxima: none, local minima: none

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(g)

\(f(x)=\frac{1}{3}x^3-9x+2\)

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Hint.

\(f'(x)=x^2-9=(x-3)(x+3)\)

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Answer.

increasing: \((-\infty,-3),(3,\infty)\text{,}\) decreasing: \((-3,3)\text{,}\) local maxima: \((-3,20)\text{,}\) local minima: \((3,-16)\)

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(h)

\(f(x)=-x^3+2x^2\)

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Hint.

\(f'(x)=-3x^2+4x=x(4-3x)\)

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Answer.

increasing: \((0,\frac{4}{3})\text{,}\) decreasing: \((-\infty,0),(\frac{4}{3},\infty)\text{,}\) local maxima: \((\frac{4}{3}, \frac{32}{27}\text{,}\) local minima: \((0,0)\)

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(i)

\(f(x)=2x^3-18x\)

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Hint.

\(f'(x)=6x^2-18=6(x-\sqrt{3})(x+\sqrt{3})\)

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Answer.

increasing: \((-\infty,-\sqrt{3}),(\sqrt{3},\infty)\text{,}\) decreasing: \((-\sqrt{3},\sqrt{3})\text{,}\) local maxima: \((-\sqrt{3}, 12\sqrt{3})\text{,}\) local minima: \((\sqrt{3}, -12\sqrt{3})\)

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Exercise Group 2.2.2. Rational and Transcendental Functions.

For each function, find any local maxima or local minima, and intervals of increase and decrease.

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(a)

\(f(x)=4\sqrt{x}-x^2+3\)

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Hint.

\(f'(x)=\frac{2}{\sqrt{x}}-2x\)

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Answer.

increasing: \((0,1)\text{,}\) decreasing: \((1,\infty)\text{,}\) local maxima: \((1,6)\text{,}\) local minima: none

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(b)

\(f(x)=\frac{x^2-3}{x-2}\text{,}\) \(x\ne 2\)

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Hint.

\(f'(x)=\frac{x^2-4x+3}{(x-2)^2}=\frac{(x-1)(x-3)}{(x-2)^2}\)

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Answer.

increasing: \((-\infty,1),(3,\infty)\text{,}\) decreasing: \((1,2),(2,3)\text{,}\) local maxima: \((1,2)\text{,}\) local minima: \((3,6)\)

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(c)

\(f(x)=x^5+x^3\)

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Hint.

\(f'(x)=5x^4+3x^2=x^2(5x^2+3)\)

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Answer.

increasing: \((-\infty,\infty)\text{,}\) decreasing: none, local maxima: none, local minima: none

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(d)

\(f(x)=x^2-2x-4\ln x\)

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Hint.

\(f'(x)=2x-2-\frac{4}{x}=\frac{2(x-2)(x+1)}{x}\)

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Answer.

increasing: \((2,\infty)\text{,}\) decreasing: \((0,2)\text{,}\) local maxima: none, local minima: \(\brac{2, -4\ln 2}\)

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(e)

\(f(x)=\frac{3}{2}x^4-x^6\)

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Hint.

\(f'(x)=6x^3-6x^5=-6x^3(x-1)(x+1)\)

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Answer.

increasing: \((-\infty,-1),(0,1)\text{,}\) decreasing: \((-1,0),(1,\infty)\text{,}\) local maxima: \(\brac{-1,\frac{1}{2}},\brac{1,\frac{1}{2}}\text{,}\) local minima: \((0,0)\)

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(f)

\(f(x)=\frac{x^2}{x-2}\)

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Hint.

\(f'(x)=\frac{x(x-4)}{(x-2)^2}\)

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Answer.

increasing: \((-\infty,0),(4,\infty)\text{,}\) decreasing: \((0,2),(2,4)\text{,}\) local maxima: \((0,0)\text{,}\) local minima: \((4,8)\)

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(g)

\(f(x)=e^{2x}+e^{-x}\)

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Hint.

\(f'(x)=2e^{2x}-e^{-x}=e^{-x}(2e^{3x}-1)\)

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Answer.

increasing: \((-\frac{\ln 2}{3},\infty)\text{,}\) decreasing: \((-\infty,-\frac{\ln 2}{3})\text{,}\) local maxima: none, local minima: \(\brac{-\frac{\ln 2}{3}, \sqrt[3]{2}+\frac{1}{\sqrt[3]{4}}}\)

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(h)

\(f(x)=x^2\ln x^2+1\)

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Hint.

\(f'(x)=2x(\ln x^2+1)\)

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Answer.

increasing: \((-\frac{1}{\sqrt{e}},0),(\frac{1}{\sqrt{e}},\infty)\text{,}\) decreasing: \((-\infty,-\frac{1}{\sqrt{e}}),(0,\frac{1}{\sqrt{e}})\text{,}\) local maxima: none, local minima: \((-\frac{1}{\sqrt{e}},1-\frac{1}{e}),(\frac{1}{\sqrt{e}},1-\frac{1}{e})\)

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(i)

\(f(x)=\frac{x^3}{3x^2+1}\)

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Hint.

\(f'(x)=\frac{3x^2(x^2+1)}{(3x^2+1)^2}\)

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Answer.

increasing: \((-\infty,\infty)\text{,}\) decreasing: none, local maxima: none, local minima: none

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(j)

\(f(x)=e^{\sqrt{x}}\)

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Hint.

\(f'(x)=\frac{e^{\sqrt{x}}}{2\sqrt{x}}\)

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Answer.

increasing: \((0,\infty)\text{,}\) decreasing: none, local maxima: none, local minima: none

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(k)

\(f(x)=x^2\ln x\)

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Hint.

\(f'(x)=2x\ln x+x=x(2\ln x+1)\)

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Answer.

increasing: \((\frac{1}{\sqrt{e}},\infty)\text{,}\) decreasing: \((0,\frac{1}{\sqrt{e}})\text{,}\) local maxima: none, local minima: \(\brac{\frac{1}{\sqrt{e}}, -\frac{1}{2e}}\)

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(l)

\(f(x)=x(\ln x)^2\)

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Hint.

\(f'(x)=(\ln x)^2+2\ln x=\ln x(\ln x+2)\)

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Answer.

increasing: \((0,\frac{1}{e^2}),(1,\infty)\text{,}\) decreasing: \((\frac{1}{e^2},1)\text{,}\) local maxima: \(\brac{\frac{1}{e^2}, \frac{4}{e^2}}\text{,}\) local minima: \((1,0)\)

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Exercise Group 2.2.3. Fractional Power Functions (Cusps and Vertical Tangents).

For each function, find any local maxima or local minima, and intervals of increase and decrease.

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(a)

\(f(x)=x^{2/3}(x^2-4)\)

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Hint.

\(f'(x)=\frac{8(x^2-1)}{3x^{1/3}}\)

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Answer.

increasing: \((-1,0),(1,\infty)\text{,}\) decreasing: \((-\infty,-1),(0,1)\text{,}\) local maxima: \((0,0)\text{,}\) local minima: \((-1,-3),(1,-3)\)

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(b)

\(f(x)=x^{1/3}(x-4)\)

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Hint.

\(f'(x)=\frac{4(x-1)}{3x^{2/3}}\)

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Answer.

increasing: \((1,\infty)\text{,}\) decreasing: \((-\infty,0),(0,1)\text{,}\) local maxima: none, local minima: \((1,-3)\)

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(c)

\(f(x)=(x-5)^{\frac{1}{3}}\)

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Hint.

\(f'(x)=\frac{1}{3(x-5)^{2/3}}\)

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Answer.

increasing: \((-\infty,5),(5,\infty)\text{,}\) decreasing: none, local maxima: none, local minima: none

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(d)

\(f(x)=(x^2-1)^{\frac{1}{3}}\)

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Hint.

\(f'(x)=\frac{2x}{3(x^2-1)^{2/3}}\)

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Answer.

increasing: \((0,1),(1,\infty)\text{,}\) decreasing: \((-\infty,-1),(-1,0)\text{,}\) local maxima: none, local minima: \((0,-1)\)

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(e)

\(f(x)=x^{\frac{1}{3}}(x+8)\)

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Hint.

\(f'(x)=\frac{4(x+2)}{3x^{2/3}}\)

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Answer.

increasing: \((-2,0),(0,\infty)\text{,}\) decreasing: \((-\infty,-2)\text{,}\) local maxima: none, local minima: \(\brac{-2,-6\sqrt[3]{2}}\)

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(f)

\(f(x)=x^{\frac{2}{3}}(x+5)\)

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Hint.

\(f'(x)=\frac{5(x+2)}{3x^{1/3}}\)

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Answer.

increasing: \((-\infty,-2),(0,\infty)\text{,}\) decreasing: \((-2,0)\text{,}\) local maxima: \(\brac{-2,3\sqrt[3]{4}}\text{,}\) local minima: \((0,0)\)

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(g)

\(f(x)=x^{\frac{1}{3}}(x^2-4)\)

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Hint.

\(f'(x)=\frac{7x^2-4}{3x^{2/3}}\)

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Answer.

increasing: \((-\infty,-\frac{2\sqrt{7}}{7}),(\frac{2\sqrt{7}}{7},\infty)\text{,}\) decreasing: \((-\frac{2\sqrt{7}}{7},0),(0,\frac{2\sqrt{7}}{7})\text{,}\) local maxima: \(\approx \brac{-\frac{2}{\sqrt{7}}, 3.12}\text{,}\) local minima: \(\approx \brac{\frac{2}{\sqrt{7}}, -3.12}\)

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(h)

\(f(x)=x^{\frac{2}{3}}(x^2-4)\)

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Hint.

\(f'(x)=\frac{8(x^2-1)}{3x^{1/3}}=\frac{8(x-1)(x+1)}{3x^{1/3}}\)

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Answer.

increasing: \((-1,0),(1,\infty)\text{,}\) decreasing: \((-\infty,-1),(0,1)\text{,}\) local maxima: \((0,0)\text{,}\) local minima: \((-1,-3),(1,-3)\)

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Exercise Group 2.2.4. Radical Functions (Square Roots).

For each function, find any local maxima or local minima, and intervals of increase and decrease.

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(a)

\(f(x)=x\sqrt{8-x^2}\)

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Hint.

\(f'(x)=\frac{-2x^2+8}{\sqrt{8-x^2}}\)

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Answer.

domain: \([-2\sqrt{2},2\sqrt{2}]\text{,}\) increasing: \((-2,2)\text{,}\) decreasing: \((-2\sqrt{2},-2),(2,2\sqrt{2})\text{,}\) local maxima: \((2,4)\text{,}\) local minima: \((-2,-4)\)

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(b)

\(f(x)=x^2\sqrt{9-x^2}\)

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Hint.

\(f'(x)=\frac{-3x^3+18x}{\sqrt{9-x^2}}\)

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Answer.

domain: \([-3,3]\text{,}\) increasing: \((-3,-\sqrt{6}),(0,\sqrt{6})\text{,}\) decreasing: \((-\sqrt{6},0),(\sqrt{6},3)\text{,}\) local maxima: \((-\sqrt{6},6\sqrt{3}),(\sqrt{6},6\sqrt{3})\text{,}\) local minima: \((0,0)\)

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(c)

\(f(x)=\sqrt{x^2-2x+2}\)

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Hint.

\(f'(x)=\frac{x-1}{\sqrt{x^2-2x+2}}\)

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Answer.

domain: \((-\infty,\infty)\text{,}\) increasing: \((1,\infty)\text{,}\) decreasing: \((-\infty,1)\text{,}\) local maxima: none, local minima: \((1,1)\)

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(d)

\(f(x)=x-6\sqrt{x-1}\)

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Hint.

\(f'(x)=1-\frac{3}{\sqrt{x-1}}=\frac{\sqrt{x-1}-3}{\sqrt{x-1}}\)

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Answer.

domain: \([1,\infty)\text{,}\) increasing: \((10,\infty)\text{,}\) decreasing: \((1,10)\text{,}\) local maxima: none, local minima: \((10,-8)\)

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(e)

\(f(x)=x^2\sqrt{5-x}\)

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Hint.

\(f'(x)=\frac{5x(4-x)}{2\sqrt{5-x}}\)

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Answer.

domain: \((-\infty,5]\text{,}\) increasing: \((0,4)\text{,}\) decreasing: \((-\infty,0),(4,5)\text{,}\) local maxima: \((4,16)\text{,}\) local minima: \((0,0)\)

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Exercise Group 2.2.5. Trigonometric Functions.

For each function, find any local maxima or local minima, and intervals of increase and decrease.

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(a)

\(f(x)=2\cos x+\sin^2 x\) on \([0,2\pi]\)

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Hint.

\(f'(x)=2\sin x(\cos x-1)\)

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Answer.

increasing: \((\pi,2\pi)\text{,}\) decreasing: \((0,\pi)\text{,}\) local maxima: \((0,2),(2\pi,2)\text{,}\) local minima: \((\pi,-2)\)

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(b)

\(f(x)=\sin x+\cos x\) on \([0,2\pi]\)

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Hint.

\(f'(x)=\cos x-\sin x\)

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Answer.

increasing: \((0,\frac{\pi}{4}),(\frac{5\pi}{4},2\pi)\text{,}\) decreasing: \((\frac{\pi}{4},\frac{5\pi}{4})\text{,}\) local maxima: \((\frac{\pi}{4},\sqrt{2})\text{,}\) local minima: \((\frac{5\pi}{4},-\sqrt{2})\)

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