Question
Asked Oct 21, 2019

Describe whether Rolle's theorem applies to the function shown below on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's theorem.

f(x)=x(x-10)^2 ; [0,10]

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Expert Answer

Step 1

Given:

Describe whether Rolle's theorem applies to the function shown below on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's theorem.

f(x)xx-10)[0,10]
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f(x)xx-10)[0,10]

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Step 2

Rolle’s Theorem:  Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f (a) = f (b), then there is at least one number c in (a, b) such that f’(c) = 0.

Step 3

Now since:

  1. f(x) is a polynomial function so it is continuous everywhere in the given interval.
  2. f’(x) is also differentiable in given open interval.
  3. f (0) = f (10) = 0.

Hence all the hypotheses are true for the given fu...

f(x)x(x-10)x(x-20x + 100)
f(x)=(x-20x+100x)
f '(x)= 3x2-40x +100
f'(c)=3c2- 40c+100
By Rolle's theorem
f'(c)= 0
3c-40c100 0
3c-30c-10c+100 0
3c(c-10)-10(c-10) 0
(c-10)(3c-10) 0
c 10
3c 10
10
c 10
3
(0,10)
But, c 10
So
10
C
3
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f(x)x(x-10)x(x-20x + 100) f(x)=(x-20x+100x) f '(x)= 3x2-40x +100 f'(c)=3c2- 40c+100 By Rolle's theorem f'(c)= 0 3c-40c100 0 3c-30c-10c+100 0 3c(c-10)-10(c-10) 0 (c-10)(3c-10) 0 c 10 3c 10 10 c 10 3 (0,10) But, c 10 So 10 C 3

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Math

Calculus