GATE 2026 DA – Question 27

Calculus and Optimization · Taylor series, maxima and minima, single-variable optimization · 1 mark · Multiple select

Let $f(x) = x^3 - 3x^2 + 2$ be a function defined on $(-1, 3]$.

Which of the following statements is/are correct?

$f(x)$ has exactly two roots in $[-0.9, 0]$.
$f(x)$ has a minimum at 2 only.
$f(x)$ has maximum at 0 only.
$f(x)$ has a root at 1.

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Correct answer: (B) $f(x)$ has a minimum at 2 only.; (D) $f(x)$ has a root at 1.

Explanation

$f(1) = 1 - 3 + 2 = 0$, so D is true, and $f(x) = (x - 1)(x^2 - 2x - 2)$ has the other roots $1 \pm \sqrt{3}$, which are $-0.73$ and $2.73$. So $[-0.9, 0]$ holds only one root and A is false. Next, $f'(x) = 3x(x - 2)$ is zero at 0 and 2. Then $f(2) = -2$ is the smallest value on the interval (as $x \to -1^+$ the function only approaches $-2$ and never reaches it, since $-1$ is excluded), so the minimum is at 2 only and B is true. The values are $f(0) = 2$ and $f(3) = 2$, so the maximum is reached at both 0 and 3 and C is false.