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Question
Verify Rolle's theorem for the function
f(x)=x2-5x+9 on [1,4]
Solution
The function f given as f(x)=x2-5x+9 is a polynomial function.
Hence
(i) it is continuous on [1,4]
(ii) differentiable on (1,4).
`Now, f(1)=1^2 - 5(1)+ 9 =1- 5+9 = 5`
`and f (4)= 4^2 - 5(4)+ 9 =16 - 20 + 9 = 5`
f (1)=f(4)
Thus, the function f satisfies all the conditions of the Rolle’s theorem.
therefore there exists c ∈ (1, 4) such that f '(c)= 0
`Now, f(x)=x^2-5x+9`
`therefore f'(x)=d/dx(x^2-5x+9)=2x-5xx1+0`
=2x-5
f'(c)=2c-5
f'(c)=0 gives, 2c-5=0
`c=5/2 in (1,4)`
Hence, the Rolle’s theorem is verified
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