GATE 2026 DA Q.27: solution and answer

GATE 2026 · DAQ.27 · 1 mark · MSQ

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

Which of the following statements is/are correct?

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

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Answer and solution

ANSWER

The correct statements are (B) and (D).

Official answer: B, DSource: Official answer key, GATE 2026 DA (IIT Guwahati)

Check (D) first. $f (1) = 1 - 3 + 2 = 0$ , so $x = 1$ is a root and (D) is true. Dividing out $(x - 1)$ gives $f (x) = (x - 1) (x^{2} - 2 x - 2)$ . Expanding confirms it: $x^{3} - 2 x^{2} - 2 x - x^{2} + 2 x + 2 = x^{3} - 3 x^{2} + 2$ .
The quadratic factor $x^{2} - 2 x - 2$ has roots $x = 1 \pm 3$ , which are about $2.732$ and $- 0.732$ . So the roots of $f$ are $- 0.732, 1, 2.732$ .
In $[- 0.9, 0]$ only $x \approx - 0.732$ lies in the interval. So $f$ has exactly one root there, not two, and (A) is false.
Critical points: $f^{'} (x) = 3 x^{2} - 6 x = 3 x (x - 2)$ , so $x = 0$ or $x = 2$ . Then $f^{'} (x) > 0$ for $x < 0$ , $f^{'} (x) < 0$ for $0 < x < 2$ and $f^{'} (x) > 0$ for $x > 2$ . Hence $f$ increases on $(- 1, 0)$ , decreases on $(0, 2)$ and increases on $(2, 3]$ .
Values: $f (0) = 2$ , $f (2) = 8 - 12 + 2 = - 2$ , $f (3) = 27 - 27 + 2 = 2$ . At the open end, $f (- 1) = - 1 - 3 + 2 = - 2$ , but $- 1$ is excluded.
Minimum: on $(- 1, 0)$ the function is increasing and tends to $- 2$ as $x \to - 1^{+}$ , so every value there is greater than $- 2$ . On $(0, 2)$ the values fall from $2$ to $- 2$ , and on $(2, 3]$ they rise from $- 2$ to $2$ . So the least value $- 2$ is attained only at $x = 2$ , and (B) is true.
Maximum: the largest value is $2$ , attained at both $x = 0$ and $x = 3$ (the point $3$ is included). So 'maximum at 0 only' is false, and (C) is false.
Correct statements: (B) and (D).

KEY IDEA

On a restricted domain, find the global extrema by comparing the critical-point values with the values at the endpoints. An endpoint can tie with a critical value, and an excluded endpoint is never attained.

WATCH OUT FOR

Taking $x = 0$ as the only maximum and ignoring that $f (3) = 2$ equals $f (0)$ because $3$ is included. A second slip is to treat $f (- 1) = - 2$ as a second minimum, although $- 1$ is excluded from the domain. A third is to count both roots $1 \pm 3$ as lying in $[- 0.9, 0]$ , when only $1 - 3 \approx - 0.732$ does.

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 5 Maxima & minima questions, with notes

GATE 2026 DA Q.27: solution and answer

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