Question

Let \( f(x) = 10x^2 + e^x - \sin(2x) - \cos x \), \( x \in \mathbb{R} \). The number of points at which the function \( f \) has a local minimum is:

• A. 0
• B. 1
• C. 2
• D. greater than or equal to 3

Hint:

To find the local minima of a function, first solve for the critical points using \( f'(x) = 0 \), then use the second derivative test to classify the critical points.

Answer 1

The Correct Option is B

Step 1: Find the first derivative of \( f(x) \).
\[ f'(x) = 20x + e^x - 2\cos(2x) + \sin x \] We set \( f'(x) = 0 \) to find the critical points. Step 2: Analyze the second derivative.
The second derivative is: \[ f''(x) = 20 + e^x + 4\sin(2x) + \cos x \] To find the points where \( f(x) \) has a local minimum, check where \( f''(x)>0 \). Step 3: Conclusion.
By solving these equations and analyzing the second derivative, we conclude that \( f(x) \) has 1 point of local minimum. Final Answer: \[ \boxed{1} \]