Math $118-$Section $3\mathrm{.}2-3\mathrm{.}4$ Worksheet

Use Calculus techniques to find the following. Show all work on scratch paper.

$f(x)=x^5-15x^3+10$

a$.$ Find all the critical points of $f(x).$

b$.$ Use the First Derivative Test to determine whether the critical points are local maxima, local minima, or neither.

c$.$ Find exact values of all inputs at which $f^{\text{'}\text{'}}(x)=0.$ Verify that these values are inflection points.

$g(x)=4x^3-5x^2-12x+6$

a$.$ Find all the critical points of $g(x).$

b$.$ Use the Second Derivative Test to determine whether the critical points are local maxima, local minima, or neither.

c$.$ Find exact values of all inputs at which $g^{\text{'}\text{'}}(x)=0.$ Verify that these values are inflection points.

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

a$.$ Find all the critical points of $g(x).$

b$.$ Use the Second Derivative Test to determine whether the critical points are local maxima, local minima, or neither.

$g(x)=\frac{1}{6}{x}^3-x^2$

a$.$ Find all the critical points of $g(x)$ and classify them as local maxima, local minima, or neither.

b$.$ Use a sign chart for $g^{\text{'}\text{'}}(x)$ to determine any inflection points. State the intervals where $g(x)$ is concave up and concave down.

c$.$ Graph the function by hand using the above points and any intercepts that you can find.

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

a$.$ Find all the critical points and use the First Derivative Test to classify them as local maxima$/$minima$/$neither$.$

b$.$ Find and verify all inflection points.

c$.$ Find the global $($absolute$)$ extrema if the function was restricted to the following intervals:

i$.-1,2$

ii$.0,2$

iii. $(-,)$

Find the $x-$value$($s$)$ at which the function has a global maximum and$/$or global minimum on the given interval.

a$.f(x)=x^3+3x^2-2$ on $-3,2$

b$.f(x)=x^2+\frac{16}{x^2}$ on $1,8$