Question

If \( f(x) = \sqrt{x} + \sin x \), then all the points of the set \( \left( x, f(x) \right)/f'(x) = 0 \) lie on:

• A. a circle
• B. a straight line
• C. an ellipse
• D. a parabola

Hint:

For functions involving square roots and trigonometric components, the equation \( f'(x) = 0 \) can often yield a parabolic relationship.

Answer 1

The Correct Option is D

We are given \( f(x) = \sqrt{x} + \sin x \). We need to find the points where \( f'(x) = 0 \). The first step is to calculate \( f'(x) \). \[ f'(x) = \frac{d}{dx} \left( \sqrt{x} + \sin x \right) = \frac{1}{2\sqrt{x}} + \cos x \] We now need to find the points where \( f'(x) = 0 \), i.e., \[ \frac{1}{2\sqrt{x}} + \cos x = 0 \] This equation implies that the points satisfying this equation lie on a parabola. The key part is the relationship between \( x \) and \( \sqrt{x} \), which is characteristic of a parabola. Hence, the points lie on a parabola.