Question

Let \( f(x) = 4x^2 - \sin x + \cos 2x \) for all \( x \in \mathbb{R} \). Determine the properties of the function \( f \):

• A. a point of local maximum
• B. no point of local minimum
• C. exactly one point of local minimum
• D. at least two points of local minima

Answer 1

The Correct Option is C

1. First Derivative: - Compute \( f'(x) \): \[ f'(x) = 8x - \cos x - 2\sin 2x. \] 2. Critical Points: - Set \( f'(x) = 0 \): \[ 8x - \cos x - 2\sin 2x = 0. \] - This equation has exactly one solution because the quadratic term \( 8x \) dominates. 3. Second Derivative: - Compute \( f''(x) \): \[ f''(x) = 8 + \sin x - 4\cos 2x. \] - Since \( f''(x) > 0 \), \( f(x) \) has a local minimum at the critical point