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GATE 2015 EC – Question 12

Engineering Mathematics · Calculus · 1 mark · Multiple choice

A function $f(x) = 1 - x^2 + x^3$ is defined in the closed interval $[-1, 1]$. The value of $x$, in the open interval $(-1, 1)$ for which the mean value theorem is satisfied, is

  1. $-1/2$
  2. $-1/3$
  3. $1/3$
  4. $1/2$

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Correct answer: (B) $-1/3$

Explanation

We have $f(1) = 1$ and $f(-1) = -1$, so the average slope is $\frac{1 - (-1)}{2} = 1$. Setting $f'(x) = -2x + 3x^2 = 1$ gives $3x^2 - 2x - 1 = 0$, whose roots are $x = 1$ and $x = -\frac{1}{3}$. Only $-\frac{1}{3}$ lies in the open interval.