Question

Consider the function $f\left(x\right)=8x^2-7x+5$ on the interval $[-6,6]$. Then the value of $c$ that satisfies the conclusion of Lagrange's mean value theorem is:

Answer 1

The function is continuous as well as diffrentiable in the given interval, since it is a quadratic polynomial.
So, LMVT can be applied on the function.
So, $f^{\prime }\left(c\right)=\frac{f\left(6\right)-f(-6)}{12}$
$\Rightarrow f^{\prime }\left(c\right)=\frac{(8\times 36-7\times 6+5)-(8\times 36+7\times 6+5)}{12}\Rightarrow f^{\prime }\left(c\right)=-7$
$\Rightarrow 16c-7=-7\Rightarrow c=0$