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GATE 2022 EC – Question 36

Engineering Mathematics · Calculus · 2 marks · Multiple choice

The function $f(x)=8\log_e x-x^2+3$ attains its minimum over the interval $[1,e]$ at $x=$ ________.

(Here $\log_e x$ is the natural logarithm of $x$.)

  1. 2
  2. 1
  3. $e$
  4. $\frac{1+e}{2}$

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Show answer and explanation

Correct answer: (B) 1

Explanation

$f'(x)=\frac8x-2x=0$ gives $x=2$, which lies in $[1,e]$ and is a maximum because $f''<0$. So the minimum occurs at an endpoint: $f(1)=0-1+3=2$ and $f(e)=8-e^2+3\approx3.61$. The minimum is at $x=1$.