Chapter 5

Find all local maximum and minimum points by the second derivative test. $$ y=\cos ^{2} x-\sin ^{2} x $$

Find all local maximum and minimum points by the second derivative test. $$ y=\sin ^{3} x $$

Describe the concavity of the functions below. $$ y=x^{2}-x $$

Describe the concavity of the functions below. $$ y=2+3 x-x^{3} $$

Describe the concavity of the functions below. $$ y=x^{3}-9 x^{2}+24 x $$

Describe the concavity of the functions below. $$ y=x^{4}-2 x^{2}+3 $$

Describe the concavity of the functions below. $$ y=3 x^{4}-4 x^{3} $$

Describe the concavity of the functions below. $$ y=\left(x^{2}-1\right) / x $$

Describe the concavity of the functions below. $$ y=3 x^{2}-\left(1 / x^{2}\right) $$

Describe the concavity of the functions below. $$ y=\sin x+\cos x $$

Describe the concavity of the functions below. $$ y=(x+1) / \sqrt{5 x^{2}+35} $$

Describe the concavity of the functions below. $$ y=x^{5}-x $$

Describe the concavity of the functions below. $$ y=6 x+\sin 3 x $$

Describe the concavity of the functions below. $$ y=x^{2}+1 / x $$

Describe the concavity of the functions below. $$ y=(x+5)^{1 / 4} $$

Describe the concavity of the functions below. $$ y=\cos ^{2} x-\sin ^{2} x $$

Describe the concavity of the functions below. $$ y=\sin ^{3} x $$

Identify the intervals on which the graph of the function \(f(x)=x^{4}-4 x^{3}+10\) is of one of these four shapes: concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing.

Describe the concavity of \(y=x^{3}+b x^{2}+c x+d\). You will need to consider different cases, depending on the values of the coefficients.

Let \(n\) be an integer greater than or equal to two, and suppose \(f\) is a polynomial of degree n. How many inflection points can \(f\) have? Hint: Use the second derivative test and the fundamental theorem of algebra.

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x^{5}-5 x^{4}+5 x^{3} $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x^{3}-3 x^{2}-9 x+5 $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=(x-1)^{2}(x+3)^{2 / 3} $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ x^{2}+x^{2} y^{2}=a^{2} y^{2}, a>0 $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x e^{x} $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=\left(e^{x}+e^{-x}\right) / 2 $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=e^{-x} \cos x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=e^{x}-\sin x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=e^{x} / x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=4 x+\sqrt{1-x} $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=(x+1) / \sqrt{5 x^{2}+35} $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x^{5}-x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=6 x+\sin 3 x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x+1 / x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x^{2}+1 / x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=(x+5)^{1 / 4} $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=\tan ^{2} x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=\cos ^{2} x-\sin ^{2} x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=\sin ^{3} x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x\left(x^{2}+1\right) $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x^{3}+6 x^{2}+9 x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x /\left(x^{2}-9\right) $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=x^{2} /\left(x^{2}+9\right) $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=2 \sqrt{x}-x $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=3 \sin (x)-\sin ^{3}(x), \text { for } x \in[0,2 \pi] $$

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. $$ y=(x-1) /\left(x^{2}\right) $$