Consider the function below.

$\displaystyle h(x) = (x + 1)^9 - 9x - 3$

(a)Find the intervals of increase.

Find the interval of decrease.

(b) Find the local minimum value.

Find the local maximum value.

(c) Find the inflection point.

Find the interval the function is concave up.

Find the interval the function is concave down.

I know that

(a) $\displaystyle (-INFINITY, -2) U (0, INFINITY)$ for interval of increase

$\displaystyle (-2, 0)$ for interval of decrease

(b) -2 for the local minimum value.

14 for the local maximum value.

(c) $\displaystyle (-1, 6)$ for the inflection point.

I got b(local minimum value) by finding the derivative which is

$\displaystyle h'(x) = 9(x + 1)^8 - 9$

then setting it equal to zero

$\displaystyle h'(x) = 9(x + 1)^8 - 9 = 0$

$\displaystyle h'(x) = 9(x + 1)^8 = 9$

$\displaystyle h'(x) = (x + 1)^8 = 1$

$\displaystyle h'(x) = x^8 = 0$

$\displaystyle x = 0$

then i plugged in 0 to the original equation

$\displaystyle h(x) = (0 + 1)^9 - 9(0) - 3$

$\displaystyle h(x) = 1 - 3$

$\displaystyle h(x) = -2$ (local minimum value)

I got b(local maximum value) by plugging in -2 to the original equation

$\displaystyle h(x) = (-2 + 1)^9 - 9(-2) - 3$

$\displaystyle h(x) = (-1) - (-18) - 3$

$\displaystyle h(x) = 17 - 3$

$\displaystyle h(x) = 14$ (local maximum value)

I got c(inflection point) by getting the 2nd derivative of the original equation

$\displaystyle h''(x) = 72(x+1)^7$

then setting it equal to zero

$\displaystyle h''(x) = 72(x+1)^7 = 0$

$\displaystyle h''(x) = (x+1)^7 = 0$

$\displaystyle h''(x) = x = -1$ (first point)

then to get the second point, i plugged in -1(first point) to the original equation

$\displaystyle h(x) = (-1 + 1)^9 - 9(-1) - 3$

$\displaystyle h(x) = 0 + 9 - 3$

$\displaystyle h(x) = 6$(second point)

My question is, is my logic correct on the way i solved the part B and the first part of part C?

also, how do i solve part A without graphing it on a calculator? is there a way to find the points without graphing it?

and how do i find the rest of part C, which is

"Find the interval the function is concave up.

Find the interval the function is concave down."

thanks!