Concave up/down and inflection points

Question

The function f(x)=x^3 - 3x^2 - 9x

D. Where is the function f(x) concave up or down?

E. What are the inflection points (x and y coordinates)?

Answer 1

When the 2nd derivative of the function is negative, the original function is concave down (think negative=frown). Similarly when positive the original is concave up (positive = smile). When the 2nd derivative is zero, that value has the potential to be the x-coordinate of a point of inflection.

f''(x)= 3x² -6x -9

f''(x) = 6x - 6

6x - 6 = 0

x = 1

Test a number to the left of 1 in the 2nd derivative.

f''(0) = -6 which is negative, so f is concave down.

Test a number to the right of 1:

f''*2) = 6, which is positive, so concave up.

Since the concavity changes at 1, there is a point of inflection there.

f(1) = -11

So there is a point of inflection at (1, -11).

Check here for confirmation:

desmos.com/calculator/4x4hxwpzuj

Answer 2

f(x) = x³ - 3x² - 9x

f'(x) = 3x² - 6x - 9

f"(x) = 6x - 6

f"(x) = 0 when x = 1

If x < 1, then f"(x) < 0. So, f is concave down.

If x > 1, then f"(x) > 0. So, f is concave up.

Since f"(1) = 0 and f"(x) changes sign at x = 1, there is an inflection point when x = 1.

Inflection point = (1, f(1)) = (1, -11)