Question
Find the absolute maximum and minimum of on the interval .
Solution — Step by Step
Setting : . Both lie inside .
This is where students slip. The absolute max/min on a closed interval occurs at either a critical point or an endpoint.
Maximum value (achieved at both and ). Minimum value (achieved at both and ).
Final answer: Absolute max , absolute min .
Why This Works
The Extreme Value Theorem guarantees that a continuous function on a closed interval attains its absolute max and min. The candidates are critical points (where or doesn’t exist) and the endpoints. Compare values at all candidates — pick the largest and smallest.
The classic mistake: assuming the maximum must be at a critical point. On a closed interval, endpoints can win. Here, ties with — both are global maxima.
Alternative Method
Use the second derivative test to classify critical points. . At , — local max. At , — local min. Then still check endpoints. The 2nd derivative test only classifies LOCAL extrema, not absolute ones.
The biggest trap: students find critical points and stop, forgetting to check endpoints. On , the endpoint achieves the max. Always evaluate at endpoints — that’s a free 2-3 marks in CBSE boards.
Common Mistake
Another subtle issue: assuming the max is at the largest critical point. Here (local max) gives , but (local min) gives . The behaviour depends on the function’s shape, not the position of the critical point on the number line.