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Mathematics - Application of Derivative Question with Solution | TestHub

MathematicsApplication of DerivativeMonotonicity-Rolle's-theroemMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

If $f (x) = sin^{2} x cos^{2} x ℓ cos^{2} x ℓ sin^{2} x ℓ sin^{2} x cos^{2} x$ , where $ℓ$ is a root of $16 ℓ^{2} - 16 ℓ + 7 = 0$ , then there exist two real numbers $(a, b) \in (0, \frac{π}{2})$ and $a < b$ for which $f (x) = 0$ . Does there exist $c \in (a, b)$ for which $f^{'} (c) = 0$ ? Then

The value of 'a' is

Options:

Answer:

Solution:

f(x) is a circulant determinant, so f(x) = (sin²x + cos²x + ℓ)(sin⁴x + cos⁴x + ℓ² − sin²x cos²x − ℓcos²x − ℓsin²x) = (ℓ+1)(ℓ² − ℓ + 1 − 3sin²x cos²x) = (ℓ+1)(ℓ² − ℓ + 1 − (3/4)sin²2x). Since 16ℓ² − 16ℓ + 7 = 0, we get ℓ² − ℓ + 1 = 9/16. So f(x) = 0 ⇒ (3/4)sin²2x = 9/16 ⇒ sin²2x = 3/4 ⇒ sin2x = √3/2 on (0, π/2), giving x = π/6 or π/3. The smaller root is a = π/6.

Stream:JEESubject:MathematicsTopic:Application of DerivativeSubtopic:Monotonicity-Rolle's-theroem

⏱ 2mℹ️ Source: QB

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