Let f be differentiable for all x . If f(1) = -2 and f'(x) ≥ 2 for x [1, 6] , then: - Mathematics MCQ AIEEE 2026 | ExamTest.live

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Let f be differentiable for all x . If f (1) = - 2 and f^{'} (x) \geq 2 for x \in [1, 6], then:

Solution & Step-by-step Explanation

Using Lagrange's Mean Value Theorem for f (x) on the interval [1, 6] : \frac{f ( 6 ) - f ( 1 )}{6 - 1} = f^{'} (c) for some c \in (1, 6) \frac{f ( 6 ) - ( - 2 )}{5} = f^{'} (c) Given f^{'} (x) \geq 2 for all x \in [1, 6], then f^{'} (c) \geq 2 . \frac{f ( 6 ) + 2}{5} \geq 2 f (6) + 2 \geq 10 f (6) \geq 8

Try it yourself before checking the explanation above.

Let f be differentiable for all x . If f (1) = - 2 and f^{'} (x) \geq 2 for x \in [1, 6], then:

f (6) \geq 8

f (6) < 8

f (6) < 5

f (6) = 5

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