GATE EC 2017 Set 2 — Question 30
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
The minimum value of the function in the interval occurs at
Correct answer
-100.01
Solution
The given function is .
To find the minimum value in the interval , we need to find the critical points by taking the first derivative and setting it to zero, and then evaluate the function at these critical points and the boundary points of the interval.
.
Both and are within the interval .
The smallest among these values is , which occurs at .Therefore, the minimum value of the function in the given interval occurs at .
To find the minimum value in the interval , we need to find the critical points by taking the first derivative and setting it to zero, and then evaluate the function at these critical points and the boundary points of the interval.
1.Find the first derivative:
.2.Find critical points:
Set :.
Both and are within the interval .
3.Evaluate the function at critical points and boundary points:
- At (critical point):
- At (critical point):
- At (boundary point):
- At (boundary point):
4.Compare the values to find the minimum:
The function values are: , , , .The smallest among these values is , which occurs at .Therefore, the minimum value of the function in the given interval occurs at .
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