At what $x$-value does a line touch $f(x)$ if $f(x)$ is tangent to the graph and parallel to a segment on the interval $[0,5]$

Question

Given $f''(x) = 3 + 4\cos(x)$, $f'(0) = 0$, and $f(0) = 0$. The line, tangent to the graph of $f(x)$ and parallel to the segment connecting the endpoints on the interval $[0,5]$, touches $f(x)$ at $x = ?$

I found $f(x) = \frac{3x^2}{2} - 4 \cos(x)$ and $f'(x) = 3x + 4 \sin(x)$. Not sure what to do next

Answer 1

Hints: 1) Your $f(x)$ is not quite correct; it doesn't satisfy the initial condition given.

2) Once you have the correct $f(x)$, can you evaluate it at $x=0, 5$ and find the slope of the segment connecting those two points?

3) That slope is the slope of the tangent line you want to find. So for what value of $x$ is the slope of the tangent line what you want?