Question

What are the critical values, if any, of `f(x) = x^5 - 5x^4 - 4x^3 - 20x^2`?

Answer 1

`x=0,4.838`

Explanation:

The critical values of a function occur whenever the function's derivative is equal to `0` or does not exist.

These are the "critical values" since they represent special points in the function--horizontal/vertical tangents, asymptotes, etc.

Thus, in order to find the function's critical values, we must first find its derivative.

Differentiation here will only require the power rule.

`f(x)=x^5-5x^4-4x^3-20x^2`

`f'(x)=5x^4-20x^3-12x^2-40x`

The derivative will never be undefined. Thus, there will only be critical values when (if) the derivative equals `0`.

`5x^4-20x^3-12x^2-40x=0`

`x(5x^3-20x^2-12x-40)=0`

From here we know one critical value is at `x=0`. As for finding when the cubic equals `0`, consult a graph since it cannot be easily factored.

The graph of `5x^3-20x^2-12x-40:`

graph{5x^3-20x^2-12x-40 [-2.5, 8, -150, 80]}

The only other critical value is when this equals `0`, at approximately `x=4.838`.

We can check a graph of the original function to see what occurs at the critical values `x=0,4.838`.

graph{x^5 - 5x^4 - 4x^3 - 20x^2 [-5, 8, -1428, 727]}

There are horizontal tangents at either location.