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Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = x3 − x2 − 12x + 7, [0, 4]

Respuesta :

Rolle's theorem works for a function [tex]f(x)[/tex] over an interval [tex][a,b][/tex] if:

  1. [tex]f(x)[/tex] is continuous on [tex][a,b][/tex]
  2. [tex]f(x)[/tex] is differentiable on [tex](a,b)[/tex]
  3. [tex]f(a)=f(b)[/tex]

This is our case: [tex]f(x)[/tex] is a polynomial, so it is continuous and differentiable everywhere, and thus in particular it is continuous and differentiable over [0,4].

Also, we have

[tex]f(0)=7=f(4)[/tex]

So, we're guaranteed that there exists at least one point [tex]c\in(a,b)[/tex] such that [tex]f'(c)=0[/tex].

Let's compute the derivative:

[tex]f'(x)=3x^2-2x-12[/tex]

And we have

[tex]f'(x)=0 \iff x= \dfrac{1\pm\sqrt{37}}{3}[/tex]

In particular, we have

[tex]\dfrac{1+\sqrt{37}}{3}\approx 2.36[/tex]

so this is the point that satisfies Rolle's theorem.

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