If c is a point at which Rolle's theorem holds for the function f(x) = _e≤ft(x^2 + α/7x) in the interval [3, 4] , then the value of c is: - Mathematics MCQ Mathematics Mock Test - 3 2026 | ExamTest.live

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If c is a point at which Rolle's theorem holds for the function f (x) = lo g_{e} (\frac{x ^{2} + α}{7 x}) in the interval [3, 4], then the value of c is:

Solution & Step-by-step Explanation

According to Rolle's theorem, f (3) = f (4) : lo g_{e} (\frac{3 ^{2} + α}{7 ( 3 )}) = lo g_{e} (\frac{4 ^{2} + α}{7 ( 4 )}) \frac{9 + α}{21} = \frac{16 + α}{28} 4 (9 + α) = 3 (16 + α) ⟹ 36 + 4 α = 48 + 3 α ⟹ α = 12 Now, f^{'} (c) = 0 . f (x) = lo g (x^{2} + 12) - lo g (7 x) = lo g (x^{2} + 12) - lo g 7 - lo g x f^{'} (x) = \frac{2 x}{x ^{2} + 12} - \frac{1}{x} = 0 \frac{2 x ^{2} - ( x ^{2} + 12 )}{x ( x ^{2} + 12 )} = 0 ⟹ x^{2} - 12 = 0 ⟹ x = 12 = 23 Since 23 \approx 3.46 lies in (3, 4), c = 23 .

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If c is a point at which Rolle's theorem holds for the function f (x) = lo g_{e} (\frac{x ^{2} + α}{7 x}) in the interval [3, 4], then the value of c is:

2

23

12

- 12

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