# Suppose that f(x) = -0.003x^4 + 0.04x^3 + 4x^2 + 3.1x + 11. Find the x-coordinates of any...

## Question:

Suppose that {eq}f(x) = -0.003x^4 + 0.04x^3 + 4x^2 + 3.1x + 11 {/eq}.

Find the {eq}x {/eq}-coordinates of any inflection points.

## Inflection Points:

We consider a continuous and differentiable function {eq}f(x). {/eq}

The inflection points of the function are obtained setting the second derivative of the function to zero, i.e. {eq}f''(x)=0. {/eq}

Given the function

{eq}f(x) = -0.003x^4 + 0.04x^3 + 4x^2 + 3.1x + 11 {/eq}

its first and second derivaitves are found as

{eq}f'(x) =-0.012x^3+0.12x^2+8x+3.1 \\ f''(x) = -0.036x^2+0.24x+8. {/eq}

The inflection points of the function are obtained setting the second derivative to zero

{eq}f''(x) =0 \Rightarrow -0.036x^2+0.24x+8 = 0 \\ \Rightarrow x=-11.94, \; \; x= 18.61. {/eq}