MonotonicityHard

Question

Suppose that f is differentiable for all x such that f’(x) ≤ 2 for all x. If f(1) = 2 and f(4) = 8 then f(2) has the value equal to -

Options

A.3
B.4
C.6
D.8

Solution


Applying LMVT in [1, 2]
= f′(c1) ∀ c1 ∈ (1,2)
f (2) - 2 ≤ 2{∵ f′(x) ≤ 2} ⇒ f(2) ≤ 4  .......(1)
Similarly applying LMVT in [2, 4]
= f′(c2) ∀ c2 ∈ (2, 4)
≤ 2 ⇒ f(2) ≥ 4    ......(2)
from (1) & (2)        f(2) = 48

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