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Verify Rolle's theorem for the function ...

Verify Rolle's theorem for the function `f(x)=x^(2)-5x+9` on [1, 4].

Text Solution

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The function f given as `f(x)=x^(2)-5x+9` is a polynomial function.
Hence it is continuous on [1, 4] and 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.
`:.` there exists `c in (1, 4)` such that f'(c)=0.
Now, `f(x)=x^(2)-5x+9`
`:.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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