Chapter 5: Curve SketchingContent check — Calculus (Guichard), Chapter 5: Curve Sketching

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=x^2-x}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=2+3x-x^3}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=x^3-9x^2+24x}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=x^4-2x^2+3}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=3x^4-4x^3}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=(x^2-1)/x}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle y=3x^2-(1/x^2)}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

$y=\cos (2x)-x$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle f(x)=\{\begin{array}{cc}x-1\hfill & x<2\hfill \\ x^2\hfill & x\ge 2\hfill \end{array}}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle f(x)=\{\begin{array}{cc}x-3\hfill & x5\hfill \end{array}}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle f(x)=x^2-98x+4}$

Find all local maximum and minimum points $(x,y)$ by the method of this section.

${\displaystyle f(x)=\{\begin{array}{cc}-2\hfill & x=0\hfill \\ 1/x^2\hfill & x\ne 0\hfill \end{array}}$

Find the local extrema of ${\displaystyle f(x)=|x|+|x-1|}$⁠.

How many critical points can a quadratic polynomial function have?

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=x^2-x}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=2+3x-x^3}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=x^3-9x^2+24x}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=x^4-2x^2+3}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=3x^4-4x^3}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=(x^2-1)/x}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle y=3x^2-(1/x^2)}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

$y=\cos (2x)-x$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle f(x)=(5-x)/(x+2)}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle f(x)=|x^2-121|}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle f(x)=x^3/(x+1)}$

Find all critical points and identify them as local maximum points, local minimum points, or neither.

${\displaystyle f(x)={\sin }^{2}x}$

Find the maxima and minima of $f(x)=\sec x$⁠.

Let ${\displaystyle f(\theta )={\cos }^2(\theta )-2\sin (\theta )}$⁠. Find the intervals where $f$ is increasing and the intervals where $f$ is decreasing in $[\mathrm{0,2}\pi ]$⁠. Use this information to classify the critical points of $f$ as either local maximums, local minimums, or neither.

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=x^2-x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=2+3x-x^3}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=x^3-9x^2+24x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=x^4-2x^2+3}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=3x^4-4x^3}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=(x^2-1)/x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=3x^2-(1/x^2)}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

$y=\cos (2x)-x$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=4x+\sqrt{1-x}}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=(x+1)/\sqrt{5x^2+35}}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=x^5-x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=6x+\sin 3x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=x+1/x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y=x^2+1/x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y={(x+5)}^{1/4}}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y={\tan }^{2}x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y={\cos }^{2}x-{\sin }^{2}x}$

Find all local maximum and minimum points by the second derivative test, when possible. (You need only find the points identified by the second derivative test; ignore any others.)

${\displaystyle y={\sin }^{3}x}$

Describe the concavity of the functions.

${\displaystyle y=x^2-x}$

Describe the concavity of the functions.

${\displaystyle y=2+3x-x^3}$

Describe the concavity of the functions.

${\displaystyle y=x^3-9x^2+24x}$

Describe the concavity of the functions.

${\displaystyle y=x^4-2x^2+3}$

Describe the concavity of the functions.

${\displaystyle y=3x^4-4x^3}$

Describe the concavity of the functions.

${\displaystyle y=(x^2-1)/x}$

Describe the concavity of the functions.

${\displaystyle y=3x^2-(1/x^2)}$

Describe the concavity of the functions.

$y=\sin x+\cos x$

Describe the concavity of the functions.

${\displaystyle y=4x+\sqrt{1-x}}$

Describe the concavity of the functions.

${\displaystyle y=(x+1)/\sqrt{5x^2+35}}$

Describe the concavity of the functions.

${\displaystyle y=x^5-x}$

Describe the concavity of the functions.

${\displaystyle y=6x+\sin 3x}$

Describe the concavity of the functions.

${\displaystyle y=x+1/x}$

Describe the concavity of the functions.

${\displaystyle y=x^2+1/x}$

Describe the concavity of the functions.

${\displaystyle y={(x+5)}^{1/4}}$

Describe the concavity of the functions.

${\displaystyle y={\tan }^{2}x}$

Describe the concavity of the functions.

${\displaystyle y={\cos }^{2}x-{\sin }^{2}x}$

Describe the concavity of the functions.

${\displaystyle y={\sin }^{3}x}$

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