GATE EE 2021 Set 1 — Question 36
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Engineering Mathematics → Calculus → Mean Value Theorems
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Question
In the open interval , the polynomial has
Correct answer
(B) one real root
Solution
Let .
Evaluate the function at the endpoints of the interval :
Since and , by the Intermediate Value Theorem, there exists at least one real root in the interval .To check for uniqueness, examine the derivative:
For , and . Therefore, for all .
Since the function is strictly decreasing in the interval , it can cross the x-axis at most once.Thus, there is exactly one real root in . Correct option is (B).
Evaluate the function at the endpoints of the interval :
Since and , by the Intermediate Value Theorem, there exists at least one real root in the interval .To check for uniqueness, examine the derivative:
For , and . Therefore, for all .
Since the function is strictly decreasing in the interval , it can cross the x-axis at most once.Thus, there is exactly one real root in . Correct option is (B).
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