GATE CS 2017 Set 2 — Question 64
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Engineering Mathematics → Calculus → Differentiation
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Question
The number of roots of in the range is
Correct answer
(C) 2
Solution
Let .
To find the number of roots, we analyze the behavior of the function:
.
To find the number of roots, we analyze the behavior of the function:
1.Derivative: .
2.Second Derivative: . Since for all , for all . This implies that is strictly increasing.
3.Critical Point: Since is strictly increasing and continuous, and while , there is exactly one root of in the interval . Let this root be . This is the global minimum of .
4.Function Values:
- At the minimum: . So the minimum value is negative.
- At the boundaries:
.
5.Conclusion: Since is continuous, starts positive at , goes below zero at , and returns to positive at , by the Intermediate Value Theorem, there are exactly two roots in the range .
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