Question

Let \(f(x)=αsin^23x\) .If  \(f ′(\dfrac{π}{12})\) = \(−3\),then the value of \(α \) is

• A. \(-1\)
• B. \(\pi\)
• C. \(\dfrac{\pi}{2}\)
• D. \(1\)
• E. \(-\pi\)

Answer 1

The Correct Option is A

Step 1: Differentiate the function \( f(x) = a\sin^2 3x \): \[ f'(x) = a \cdot 2\sin 3x \cdot \cos 3x \cdot 3 = 6a \sin 3x \cos 3x \] Using the double-angle identity: \[ f'(x) = 3a \sin 6x \]

Step 2: Evaluate the derivative at \( x = \frac{\pi}{12} \): \[ f'\left(\frac{\pi}{12}\right) = 3a \sin\left(6 \cdot \frac{\pi}{12}\right) = 3a \sin\left(\frac{\pi}{2}\right) = 3a \cdot 1 = 3a \]

Step 3: Set the derivative equal to the given value: \[ 3a = -3 \] \[ a = -1 \]

Answer 2

Given

\(f(x) = α sin^23\)

derivate both side with respect to \(x\)\(\)

\(\therefore f' (x) = 3α sin 6x\)

\(f ′ (\dfrac{π}{12}) = −3\)          \(\therefore\) (\(Sin(\dfrac{\pi}{2}=1)\))

\(3α sin(\dfrac{π}{2}) = −3\)

\(α = −1\)