Verify Rolle’s theorem for the function $f(x) = {x^2} + 2x - 8,x[ - 4,2]$ .

Consider, $f(x) = {x^2} + 2x - 8\,in\,[ - 4,2]$ which isa polynomial function and so

(i) Functionf(x) is continuous in [$-$4, 2]

(ii) f(x) is derivable in ($-$4, 2) and

(iii) f($-$4) $=$ 0 and f(2) $=$ 0 $\Rightarrow$ f($-$4) $=$f(2).

Hence,conditions of Rolle's theorem are satisfied. Hence there exists, at least one $c \in ( - 4,2)$such that $f( - 4) = f(2).$

Now, $f'(c) = 0 \Rightarrow 2c + 2 = 0 \Rightarrow c = - 1 \in ( - 4,2).$ Thus, $- 1 \in ( - 4,2)$such that $f'( - 1) = 0$. Hence, Rolle’s theorem is verified.