I need help understanding this question: MAX and MIN problem

Question

Find all critical numbers of the polynomialf(x)=6x^3+9x^2−36x−4and then list them (separated by commas) in the box below. If there are no critical numbers, enter None

Answer 1

Definition A critical number of a function f is a number x₀ in the domain of f such that

either f'(x₀) does not exist or f'(x₀) = 0

where f'(x₀) means the derivative of f at x₀

Example Let's see an example with a polynomial slightly different from yours so that you can solve yours on your own. Let's find all critical numbers of the given polynomial:

g(x) = x³-3x+10

Solution

To find all its critical numbers we need to find all values of x for which the derivative of g either does not exist or if the derivative exists at x, the derivative at x is equal to zero.

Since g is a polynomial, it is always differentiable (its derivative exists everywhere), thus, to find g's critical numbers, we only need to find the values of x such that g'(x)=0.

Let's compute g'(x):

g'(x) = 3x²-3

Hence, g'(x) = 0 if and only if 3x²-3 = 0.

We solve the equation: 3x²-3 = 0 => 3(x²-1)=0 => x=+-1

It follows that the critical numbers of g are -1 and 1 (please note this is the solution for a modified question).

Now, your turn. Good luck!

Bonus questions:

1. Given a polynomial g of degree 3, what is the maximum number of critical numbers that g can have?
2. What would happen if we had g(x) = x³+3x+10? How many critical numbers would g now have?

Answer 2

f'(x) = 18x² + 18x - 36 = 0; x² + x - 2 = 0; (x + 2)(x - 1) = 0; x = - 2 and x = 1 are 2 critical numbers;

f''(x) = 2x + 1; f''(- 2) = - 3 < 0; at x = - 2 we have max

f''(1) = 1) = 3 > 0; at x = 1 we have min