$10$ pts $4)$ A certain function $f$ is differentiable and its derivative is $f^{\text{'}}(x)=10x^2+11x-8.$

i$)$ Solve the equation $f^{\text{'}}(x)=0.$

The solutions are

ii$)$ Find the general antiderivative of $f^{\text{'}}(x)=10x^2+11x-8$

iii$)$ Derive the explicit formula for $f(x),$ given that $f(3)=2\mathrm{.}5.$ Show your work to earn credit.

$f(x)=$

$$

iv$)$ List all the intervals over which the function $f$ is increasing: $$

v$)$ List all the intervals over which the function $f$ is decreasing: $$

vi$)$ This function $f$ has a local maximum at a point $x=c.$ Calculate $f(c).$

vii$)$ This function $f$ has a local minimum at a point $x=k.$ Calculate $f(k).$

viii$)$ Calculate the second derivative of $f.f^{\text{'}\text{'}}(x)=$

ix$)$ Find all the inflection points of $f.$ For full credit, you must also make good a sign chart for $"\text{'}.$

Give both coordinates for each inflection point. $$I