GATE DA 2026 Set 1 — Question 27
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Calculus & Optimization → Maxima, Minima & Optimization → Maxima & Minima
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Question
Let be a function defined on .Which of the following statements is/are correct?
Correct answer
(B) f(x) has a minimum at 2 only.; (D) f(x) has a root at 1.
Solution
Given on .1. Find Roots:
. So is a root. (Option D is TRUE).
Divide by :
.
Roots of are .
, , .
Roots in : Only . There is exactly one root in this interval. (Option A is FALSE).2. Find Extrema:
.
Critical points at and .
Evaluate at critical points and endpoints:
. So is a root. (Option D is TRUE).
Divide by :
.
Roots of are .
, , .
Roots in : Only . There is exactly one root in this interval. (Option A is FALSE).2. Find Extrema:
.
Critical points at and .
Evaluate at critical points and endpoints:
- : (Open interval, value not attained).
- : (Local Max).
- : (Local Min).
- : (Endpoint Max).
- Option B: Minimum value is . It is attained at . Is it attained elsewhere? Solve . . Roots are . Since is not in the domain , the minimum is attained only at . (Option B is TRUE).
- Option C: Maximum value is . It is attained at and . So it is not at 0 "only". (Option C is FALSE).
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