Question
Find the maximum value of on the interval . (CBSE 2023 4-mark PYQ)
Solution — Step by Step
Critical points: and . Both lie inside the interval , so they’re candidates for the maximum.
Among , the maximum is at .
Final answer: Maximum value is , attained at .
Why This Works
For continuous functions on a closed interval, the absolute extrema occur either at critical points (where ) or at the endpoints. Always check all four candidates — never assume the critical point gives the maximum.
CBSE board examiners deliberately design intervals where the maximum is at an endpoint, not at a critical point. Students who skip endpoints lose 2 marks.
Alternative Method
Test the second derivative at each critical point: . (local max), (local min). Local max at gives , but absolute max is still found by comparing with endpoints.
The closed-interval method works in 3 steps: find , list candidates ( inside + endpoints), evaluate at each. Don’t overthink — JEE Main 2024 had this exact pattern.
Common Mistake
Stopping at the local maximum and reporting as the answer. The interval extends to , where is much larger. Always evaluate at endpoints.