1. ## Exponential function

How to complete the following table?

And also:
a) f(x) = 2x^3 + 3x^2 - 12x + 6 has critical points at x = -2 and x = 1. Use the second derivative test to determine whether these are local maximum or local minimum values.

b)
how to Calculate f(-2) and f(1) . Briefly explain the appropriateness of these two values

2. Originally Posted by Snowboarder
How to complete the following table?

And also:
a) f(x) = 2x^3 + 3x^2 - 12x + 6 has critical points at x = -2 and x = 1. Use the second derivative test to determine whether these are local maximum or local minimum values.

b) how to Calculate f(-2) and f(1) . Briefly explain the appropriateness of these two values
$\displaystyle f(x) = 2x^3 + 3x^2 - 12x + 6$

taking the derivative we get

$\displaystyle f'(x)=6x^2+6x-12=6(x^2+x-2)=6(x+2)(x-1)$

So our critical points are -2 and 1

now taking the 2nd derivative we get

$\displaystyle f''(x)=12x+6$

Now we check the above critical points

$\displaystyle f''(-2)=12(-2)+6=-18$ so this is a max

$\displaystyle f''(1)=12(1)+6=18$ so this is a min.

Good luck.