GATE EC 2019 Set 1 — Question 36
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Engineering Mathematics → Calculus → Mean Value Theorems
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Question
Consider a differentiable function on the set of real numbers such that and . Given these conditions, which one of the following inequalities is necessarily true for all ?
Correct answer
(B) f(x) ≤ 2 x + 1
Solution
By the Mean Value Theorem, for any , there exists a between and such that:Given , we have:Taking the absolute value on both sides:Since for all , we have . Thus:Since , it follows that is necessarily true for all .
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