Rolle's Theorem

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Wize University Calculus 1 Textbook > Applications of Differentiation

Rolle's Theorem

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The function f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​ has no horizontal tangent lines on the interval [−1,1][-1,1][−1,1] , even though f(−1)=f(1)f(-1)=f(1)f(−1)=f(1). Why doesn't this contradict Rolle's Theorem?

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More Rolle's Theorem Questions:

Prove that f(x)=x3−6x2+11x+5f(x)=x^3-6x^2+11x+5 f(x)=x3−6x2+11x+5 has exactly one zero on the interval [−1,1][-1,1][−1,1]. 

Practice: Exactly One Root

Q.\textbf{Q.}Q. Show that the equation x+sin⁡x=1x+\sin{x}=1x+sinx=1 has exactly one solution in the interval [0,1][0,1][0,1]. 

Rolle's Theorem: Exactly One Solution

Show that the equation cos⁡2(x)=2x\cos^2(x)=2xcos2(x)=2x has exactly one solution on R\mathbb{R}R.