GATE EC 2014 Set 3 — Question 36
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
The maximum value of in the interval is __________
Correct answer
5.9 to 6.1
Solution
To find the maximum value of a continuous function on a closed interval , we need to evaluate the function at its critical points within the interval and at the endpoints of the interval. The largest of these values will be the absolute maximum.The given function is on the interval .Step 1: Find the critical points.
Critical points occur where the first derivative is zero or undefined. Since is a polynomial, its derivative is defined everywhere.
Set to find the critical points:
Divide by 6:
Factor the quadratic equation:
The critical points are and . Both of these points lie within the interval .Step 2: Evaluate the function at the critical points and the endpoints.
The points to check are the endpoints and the critical points .
The values of the function at the points of interest are -3, 2, 1, and 6.The maximum value among these is 6.Thus, the maximum value of the function in the interval is 6.
Critical points occur where the first derivative is zero or undefined. Since is a polynomial, its derivative is defined everywhere.
Set to find the critical points:
Divide by 6:
Factor the quadratic equation:
The critical points are and . Both of these points lie within the interval .Step 2: Evaluate the function at the critical points and the endpoints.
The points to check are the endpoints and the critical points .
- At endpoint :
- At critical point :
- At critical point :
- At endpoint :
The values of the function at the points of interest are -3, 2, 1, and 6.The maximum value among these is 6.Thus, the maximum value of the function in the interval is 6.
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