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GATE 2025 EC – Question 51

Engineering Mathematics · Calculus · 2 marks · Multiple select

Consider a non-negative function $f(x)$ which is continuous and bounded over the interval $[2,8]$. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$ over the interval.

Among the combinations of $\alpha$ and $\beta$ given below, choose the one(s) for which the inequality

$$\beta\le\int_2^8f(x)\,dx\le\alpha$$

is guaranteed to hold.

  1. $\beta=5m,\ \alpha=7M$
  2. $\beta=6m,\ \alpha=5M$
  3. $\beta=7m,\ \alpha=6M$
  4. $\beta=7m,\ \alpha=5M$

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Correct answer: (A) $\beta=5m,\ \alpha=7M$

Explanation

Over an interval of length 6, $6m\le\int_2^8f\,dx\le6M$. Since $f\ge0$ we have $m,M\ge0$, so $5m\le6m$ and $7M\ge6M$: option A always holds. In B, $5M<6M$ can fail (e.g. constant $f$ gives $\int=6M>5M$). In C and D, $7m>6m$ can fail the lower bound when $m>0$.