What is the maximum number of relative extrema contained in the graph of this function? f(x)=3x^3-x^2+4x-2

Given the function:

f(x) = 3x^3 - x^2 + 4x - 2

Find the relative extrema.

Solution:

In order to find the relative extrema of the given function, we must first take its first derivating. We simply need to apply power rule in our case since we have a cubic function. 

f(x) = 3x^3 - x^2 + 4x - 2

f'(x) = 9x^2 - 2x + 4

Next find the value of x.

In order to find the value of x, we simply needed to apply the quadratic equation.

a = 9

b = -2

c = 4

The Quadratic formula:

x = −b ± √(b^2 − 4ac)/2a

is used to solve quadratic equations where a ≠ 0, in the form
ax^2+bx+c=0

When b^2−4ac=0 there is one real root.

When b^2−4ac>0 there are two real roots.

When b^2−4ac<0 there are no real roots, only a complex number.

Substitute the given values of a, b and c to the quadratic formula.

x = −b ± √(b^2 − 4ac)/2a

x = 2 ± √(4 - 144)/-4

x = 2 ± √(-140)/-4

Since we have a negative value in our square root sign, then it means that the answer will be an imaginary number, or a complex number. Hence, due to this fact, we do not have any relative extrema in the function.