Question

Let \[ f(x) = \begin{cases} \frac{1 - \sin^3 x}{3 \cos^2 x}, & x<\frac{\pi}{2} \\ \alpha, & x = \frac{\pi}{2} \\ \frac{\beta(1 - \sin x)}{(\pi - 2x)^2}, & x>\frac{\pi}{2} \end{cases} \] If \( f(x) \) is continuous at \( x = \frac{\pi}{2} \), find \( \alpha \beta \).

• A. 1
• B. -1
• C. 2
• D. -2

Hint:

Use Taylor expansion around critical points to evaluate one-sided limits involving trigonometric functions.

Answer 1

The Correct Option is C

To ensure \( f(x) \) is continuous at \( x = \frac{\pi}{2} \), the left-hand limit, right-hand limit, and the value of the function at \( x = \frac{\pi}{2} \) must be equal. Let's calculate these limits:
Left-hand limit as \( x \to \frac{\pi}{2}^- \):The function is given by\( f(x) = \frac{1 - \sin^3 x}{3 \cos^2 x} \).
As \( x \to \frac{\pi}{2} \), \(\sin x \to 1\) and \(\cos x \to 0\).
Performing a Taylor expansion of \(\sin x\) around \(\pi/2\): 
\(\sin x \approx 1 - \frac{1}{2}(x - \frac{\pi}{2})^2 = 1 - \frac{y^2}{2}\) where \((y = x - \frac{\pi}{2})\).
\(\cos x \approx y\).
\(1 - \sin^3 x \approx 1 - (1 - \frac{3y^2}{2} + \frac{y^6}{8}) \approx \frac{3y^2}{2}\).
\(3 \cos^2 x \approx 3y^2\).
\(\Rightarrow \frac{1 - \sin^3 x}{3 \cos^2 x} \approx \frac{3y^2/2}{3y^2} = \frac{1}{2}\).
Thus, the left-hand limit is \(\frac{1}{2}\).
Right-hand limit as \( x \to \frac{\pi}{2}^+ \):
The function is given by \( f(x) = \frac{\beta (1-\sin x)}{(\pi - 2x)^2} \).
As \( x \to \frac{\pi}{2} \), \(\sin x \to 1\).
Using Taylor expansion: \(1 - \sin x \approx \frac{1}{2}(x-\frac{\pi}{2})^2\).
Let \( z = \pi - 2x \), then \( (x-\frac{\pi}{2}) = -\frac{z}{2} \),
\(1 - \sin x \approx \frac{z^2}{8} \).
\(\Rightarrow \frac{\beta(1-\sin x)}{z^2} \approx \frac{\beta z^2/8}{z^2} = \frac{\beta}{8}\).
Thus, the right-hand limit is \(\frac{\beta}{8}\).
For continuity at \( x = \frac{\pi}{2} \), left-hand limit = right-hand limit = \( \alpha \).
\(\frac{1}{2} = \alpha = \frac{\beta}{8}\).
This implies \(\beta=4\) and \(\alpha=\frac{1}{2}\).
Finally, \(\alpha \beta = \frac{1}{2} \times 4 = 2\).