Find all numbers c that satisfy the conclusion of the Mean Value Theorem for the following function and interval. Enter the values in increasing order and enter N in any blanks you don't need to use. f(x)=24x2+16x+7[−11]

Answer:[tex]c = \frac{-2}{3}[/tex]Step-by-step explanation:We are given the following:[tex]f(x)=24x^2+16x+7[/tex]Interval: [-1,1]Mean Value theorem:If the given function is continuous n [a,b] and differentiable in (a,b), then there exist a c in ((a,b) such that[tex]f'(c) = \displaystyle\frac{f(b)-f(a)}{b-a}[/tex]Now, we evaluate f(-1), f(1) and f'(x)[tex]f'(x) = 48x+16\\f(1)= 24(1)^2+16(1)+7 = 47\\f(-1) = 24(-1)^2+16(-1)+7 = 15[/tex]By mean value theorem:[tex]f'(c) = \displaystyle\frac{f(b)-f(a)}{b-a}\\\\48c + 16 = \frac{15-47}{1-(-1)}\\\\48c + 16 = -16\\48c = -32\\\\c = \frac{-32}{48} = \frac{-2}{3}\\\\[/tex]