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Verify Mean Value Theorem, if f(x)=x^3-5...

Verify Mean Value Theorem, if `f(x)=x^3-5x^2-3x` in the interval [a, b], where `a = 1 and b = 3`. Find all `c in (1, 3)` for which `f^(prime)(c)=0`.

Text Solution

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Mean value theorem states that if a function `f(x)` is continuous and differentiable in interval `(a,b)`, then,
`(f(b)-f(a))/(b-a) = f'(c)`, where c lies in `(a,b)`.
Here, `f(x) = x^3-5x^2-3x`
`:. f'(x) = 3x^2-10x-3`
Here, we are given, `b = 3 and a = 1`
`:. f(b) = f(3) = 3^3 - 5(3)^2-3(3) = 27-45-9 = -27`
`f(a) = f(1) = 1^3 - 5(1)^2-3(1) = 1-5-3 = -7`
`:. (f(b)-f(a))/(b-a) = (-27+7)/(3-1) = -10`
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