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Mathematics Question on Differentiability

Let f′(x),f ' (x), be differentiable ∀ x.\forall \, x. If f(1)=−2f (1) = -2 and f′(x)≥2∀x∈[1,6],f '(x) \geq 2 \forall x \in [1, 6], then

A

f(6)<8f (6) < 8

B

f(6)≥8f (6) \geq 8

C

f(6)≥5f (6) \geq 5

D

f(6)≤8f (6) \leq 8

Answer

f(6)≥8f (6) \geq 8

Explanation

Solution

Since, f′(x)f' (x) is differentiable ∀x∈[1,6]\forall x \in [1, 6]
∴\therefore By Lagrange?s mean value theorem,
f′(x)=f(6)−f(1)6−1f'(x) = \frac{f(6) - f(1)}{6 - 1}
∵f′(x)≥2∀x∈[1,6]\because f'(x) \geq 2 \forall x \in [1, 6 ] (given )
⇒ ∴f(6)+25≥2\Rightarrow \,\therefore \frac{f(6) + 2 }{5} \geq 2 [∵ f(1)=−2\because \, f(1) = - 2]
⇒f(6)≥10−2 ⇒f(6)≥8\Rightarrow f(6) \geq 10 - 2 \, \Rightarrow f(6) \geq 8