Question

If a continuous, differentiable function f has values f (−2) = 3 and f (4) = 1, what can you say about f ' on [−2, 4]?

Short answer

f is continuous and differentiable on $\left[-2,4\right]$and satisfied all conditions of Rolle's theorem .

Step-by-step solution

Step 1. Given information .

Consider the given points of function f$\left[-2,4\right]$.

Step 2. Find function f to satisfy the given conditions .

$f\left(x\right)=\left(x+2\right)\left(x-4\right)$has roots $x=-2,x=4$.

The function that satisfied the conditions of Rolle's theorem is $f\left(x\right)=x^2-2x-8$Since f is a polynomial, it is continuous and differentiable. In particular, it is continuous on $\left[-2,4\right]$.

Therefore Rolle’s Theorem applies to the function f , and we can conclude that there must exist some value of c ∈ (−2, 4) for which f '(c) = 0. At this value f will have a horizontal tangent line .