Q. 54 - Calculus

Question

Determine whether or not each function f in Exercises 53–60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.  

$f\left(x\right)=x^3-2x+1,[a,b]=[0,6]$

Step-by-Step Solution

Answer

The function satisfies the mean value theorem and $c=\pm 2\sqrt{3}$

Step 1. Given information

$f\left(x\right)=x^3-2x+1$

Step 2. Proving Mean Value Theorem

$\begin{array}{r}{f}^{\mathrm{\prime }}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\\ =\frac{6^3-2\cdot 6+1-\left(0^3-2\cdot 0+1\right)}{6-0}\\ =\frac{216-12+1-1}{6}\\ =34\end{array}$

Now,

$\begin{array}{r}{f}^{\mathrm{\prime }}\left(c\right)=34\\ 3c^2-2=34\\ 3c^2=36\\ c^2=12\\ c=\pm \sqrt{12}\\ c=\pm 2\sqrt{3}\end{array}$