GATE CE 2022 Set 1 — Question 45
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Engineering Mathematics → Calculus → Differentiability
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Question
Let denote the maximum of two real numbers and . Which of the following statement(s) is/are TRUE about the function ?
Correct answer
(A) It is continuous on its domain.; (B) It has a local minimum at x = 2.
Solution
The function is defined as . To find the intersection point:
.For , , so .
For , , so .Thus,
.For , , so .
For , , so .Thus,
- Continuity: At , and . Since the limits match the function value , the function is continuous. (Statement A is TRUE).
- Extrema: The graph of the function is a V-shape with its vertex at . The function decreases for and increases for . Therefore, it has a local minimum at . (Statement B is TRUE, Statement C is FALSE).
- Differentiability: The left-hand derivative at is , and the right-hand derivative is . Since they are not equal, the function is not differentiable at . (Statement D is FALSE).
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