GATE EC 2025 Set 1 — Question 51
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Engineering Mathematics → Calculus → Definite & Improper Integrals
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Question
Consider a non-negative function which is continuous and bounded over the interval . Let and denote, respectively, the maximum and the minimum values of over the interval. Among the combinations of and given below, choose the one(s) for which the inequality is guaranteed to hold.
Correct answer
(A) β = 5m, α = 7M
Solution
For a continuous function on the interval , the property of definite integrals states:where and are the minimum and maximum values of on .Here, and , so the interval length is . Thus:For the inequality to be guaranteed for any such non-negative function, we must satisfy:
and .Since , we have and .(A) and . This is always true for non-negative .
(B) is true, but is false for .
(C) is false for , but is true.
(D) Both conditions fail for .Therefore, only combination (A) is guaranteed to hold.
and .Since , we have and .(A) and . This is always true for non-negative .
(B) is true, but is false for .
(C) is false for , but is true.
(D) Both conditions fail for .Therefore, only combination (A) is guaranteed to hold.
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