Question

If function \( f \) is continuous at point \( x = \pi \) and \[ f(x) = \begin{cases} kx + 1, & x \le \pi \\ \cos x, & x>\pi \end{cases} \] then the value of \( k \) is ______

• A. \( \frac{2}{\pi} \)
• B. \( -\frac{2}{\pi} \)
• C. \( \frac{1}{\pi} \)
• D. \( 0 \)

Hint:

For piecewise functions, always equate LHL and RHL at the boundary point.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a function to be continuous at $x = a$, the Left Hand Limit (LHL) must equal the Right Hand Limit (RHL) and the value of the function $f(a)$.
Step 2: Formula Application:
LHL (at $x = \pi$): $\lim_{x \to \pi^-} (kx + 1) = k\pi + 1$. RHL (at $x = \pi$): $\lim_{x \to \pi^+} \cos x = \cos \pi = -1$.
Step 3: Explanation:
Since $f$ is continuous: $k\pi + 1 = -1$ $k\pi = -2 \implies k = -2/\pi$.
Step 4: Final Answer:
The value of $k$ is $-2/\pi$.