For each function,

(a) find $y^{\prime}=f^{\prime}(x)$.

(b) find the critical values.

( c ) find the critical points.

(d) find intervals of $x$-values where the function is increasing and where it is decreasing.

(e) classify the critical points as relative maxima, relative minima, or horizontal points of inflection. In each case, check your conclusions with a graphing calculator.

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

Find the x-coordinates of all critical points of the given function. Determine whether each critical point is a relative maximum, minimum, or neither by first applying the second derivative test, and, if the test fails, by some other method. $f(x)=x e^{-x^{2}}$

In the said exercise, you found the critical points of the competing-species system. Assume the critical point given by $\left(\frac{a n-m c}{b n-c p}, \frac{b m-a p}{b n-c p}\right)$ is the initial condition and repeat the said exercise. Compare the results.

Use the given derivative to find all critical points of f, and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. Assume in each case that f is continuous everywhere.

$$f'(x)=e^{2x}-5e^x+6$$

Find the critical points of the functions. For each critical point, determine, by the second-derivative test, whether it corresponds to a relative maximum, to a relative minimum, or to neither, or whether the test gives no information. $f(x, y)=x^{2}+4 y^{2}-6 x-32 y+1$

For each function,

(a) find $y^{\prime}=f^{\prime}(x)$.

(b) find the critical values.

( c ) find the critical points.

(d) find intervals of $x$-values where the function is increasing and where it is decreasing.

(e) classify the critical points as relative maxima, relative minima, or horizontal points of inflection. In each case, check your conclusions with a graphing calculator.

$$y=\frac{x^3}{3}+\frac{x^2}{2}-2 x+1$$

Find the critical points of the following function. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. $f(x, y)=\sin (2 \pi x) \cos (\pi y), \text { for }|x| \leq \frac{1}{2} \text { and }|y| \leq \frac{1}{2}$

Find the critical points of the functions. For each critical point, determine, by the second-derivative test whether it corresponds to a relative maximum, to a relative minimum, or to neither, ot whether the test gives no information. $f(p, q)=p q-\frac{1}{p}-\frac{1}{q}$

For each abbreviation below add periods where necessary.

**Example **1, Prot C. S. Blyth

5. Dr Mary Frances

Two trigonometric functions f and g have periods of 2, and their graphs intersect at x = 5.35. Determine one negative value of x at which the graphs intersect.

Find the critical numbers of the following function.

$$G(x)=\sqrt[3]{x^2-x}$$

Sketch the graph of the function. (Include two full periods.) Use a graphing utility to verify your result.

$$y=-4 \tan \frac{x}{3}$$

Graph three periods of the function. Use your understanding of transformations, not your graphing calculators. Be sure to show the scale on both axes.

$$y=8cos5x$$

Find the critical numbers of the following function.

$$f(x)=5+6 x-2 x^3$$