List the critical point(s) of f

Question

Let f(x)=x^3+9x^2−21x+4

Part 1: Critical Points

List the critical point(s) of f

Answer 1

Critical points are where the first derivative = 0 or is undefined.

f(x) = x^3+9x^2−21x+4

f'(x) = 3x^2 +18x - 21 is defined for all values

3x^2 + 18x - 21 = 0

x^2 +6x -7 = 0

(x+7)(x-1) = 0

x = -7 and 1

Answer 2

Critical Points are where the slope of the function is zero, and the point is in the domain of the function.

Aka f'(x) = 0

Take the derivative of f(x)

f'(x) = 3x²+18x-21

Find the roots

3x²+18x-21 = 0 -> x²+6x - 7 = 0 -> (x+7)(x-1)=0

Thus

x = 1,-7

since the domain is all real numbers

The critical points are 1 and -7