local minimum or local maximum

Question

Given the function g(x)=6x3+9x2−36x, find the first derivative, g'(x).

g'(x)= 

 Notice that g'(x)=0 when x=1, that is, g'(1)=0.

Now, we want to know whether there is a local minimum or local maximum at x=1,

so we will use the second derivative test.

Find the second derivative, g''(x)

.g''(x)=

  Evaluate g''(1).

g''(1)=

Based on the sign of this number, does this mean the graph of g(x) is concave up or concave down at x=1

At x=1 the graph of g(x) is?= concave up or concave down?

 Based on the concavity of g(x) at x=1, does this mean that there is a local minimum or local maximum at x=1?

At x=1 there is a local=concave up or concave down?

Answer 1

Let g(x) = 6x³ + 9x² - 36x. Then, the first derivative is g'(x) = 18x² + 18x - 36 and we can confirm that

g'(1) = 18(1)² + 18(1) - 36 = 0

The second derivative is g''(x) = 36x + 18 such that g''(1) = 36(1) + 18 = 54. Since g''(1) > 0, we say that at x = 1, g(x) is concave up, implying that there is a local minimum.