Question

Let \(f(x) = \begin{cases} x + \alpha, & \text{if } x < 0 \\ \max(2\cos x, 2\sin x), & \text{if } x \geq 0 \end{cases}\). If \(f\) is continuous at \(x = 0\), then the value of \(\alpha\) is equal to

• A. 2
• B. 3
• C. 4
• D. 1
• E. 0

Hint:

To find \(\max\) of two functions near a point, compare their values at that point.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For continuity at \(x = 0\), LHL = RHL = \(f(0)\).

Step 2: Detailed Explanation:
At $x = 0$, $2\cos 0 = 2$ and $2\sin 0 = 0$. For $x>0$ and very small, $\cos x>\sin x$, so $\max(2\cos x, 2\sin x) = 2\cos x$.
RHL = $\lim_{x \to 0^+} 2\cos x = 2$.
LHL = $\lim_{x \to 0^-} (x + \alpha) = \alpha$.
For continuity: $\alpha = 2$.

Step 3: Final Answer:
\(\alpha = 2\).