Question

Let the function $ f(x) $ be defined as follows: $$ f(x) = \begin{cases} (1 + | \sin x |)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}<x<0 \\b, & x = 0 \\ \frac{\tan 2x}{\tan 3x}, & 0<x<\frac{\pi}{6} \end{cases} $$ Then the values of $ a $ and $ b $ are:

• A. \( a = -\frac{2}{3}, b = \frac{2}{3} \)
• B. \( a = \frac{2}{3}, b = e^{\frac{2}{3}} \)
• C. \( a = e^{\frac{2}{3}}, b = \frac{2}{3} \)
• D. \( a = \frac{2}{3}, b = e^{-\frac{2}{3}} \)

Hint:

For piecewise functions, ensure continuity at the transition points by matching the limits and values at \( x = 0 \).

Answer 1

The Correct Option is B

We are given a piecewise function. First, we analyze the behavior of the function at different intervals:
Step 1: For \( -\frac{\pi}{6} < x < 0 \),
\[ f(x) = (1 + |\sin x|)^{\frac{a}{|\sin x|}}. \]
Step 2: At \( x = 0 \),
\[ f(x) = b. \] 
Step 3: For \( 0 < x < \frac{\pi}{6} \),
\[ f(x) = \frac{\tan 2x}{\tan 3x}. \] We then compare the limits and continuity at each interval to determine the correct values of \( a \) and \( b \).