# Particle Motion/Mean Value Theorem

Here's a two-part question: "Consider the function $f(x)=2x^3−9x^2−24x+1$ on the interval $[−6,8]$. Find the average or mean slope of the function on this interval."

What I did in my initial attempt was derive the first function, which I assumed to be $s(t)$, and then set slope equal to $v(8)-v(-6)\over14$. This came out to $m=-6$, which was incorrect.

"By the Mean Value Theorem, we know there exists a $c$ in the open interval $(−6,8)$ such that $f′(c)$ is equal to this mean slope. For this problem, there are two values of c that work."

For this part, I set $v(t)$ equal to -6, and solved for $t$. The values were obviously not correct. Where did I go wrong?

## 1 Answer

Mean slope on the interval $[a,b]$ is defined by $m=\frac{f(b)-f(a)}{b-a}$ and not $\frac{f^\prime(b)-f^\prime(a)}{b-a}$. so $m$ should be 62 ...