Question

If \(f(x) =   \begin{cases}  2x & \text{for}\ x\lt1 \\  5a-x & \text{for}\ x\geq1 \end{cases}\) is continuous on \(\R\), then the value of a is equal to

• A. \(\frac{1}{5}\)
• B. \(\frac{2}{5}\)
• C. \(\frac{3}{5}\)
• D. \(\frac{4}{5}\)
• E. 1

Answer 1

The Correct Option is C

We are given the piecewise function:

\[ f(x) = \begin{cases} 2x & \text{for} \ x < 1 \\ 5a - x & \text{for} \ x \geq 1 \end{cases} \]

For \( f(x) \) to be continuous at \( x = 1 \), the values of the function from both sides must be equal at \( x = 1 \).

For \( x \to 1^- \) (as \( x \) approaches 1 from the left):

\[ \lim_{x \to 1^-} f(x) = 2(1) = 2 \]

For \( x \to 1^+ \) (as \( x \) approaches 1 from the right):

\[ \lim_{x \to 1^+} f(x) = 5a - 1 \]

For continuity at \( x = 1 \), we equate both limits:

\[ 2 = 5a - 1 \]

Solving for \( a \):

\[ 2 + 1 = 5a \] \[ 3 = 5a \] \[ a = \frac{3}{5} \]

Answer: \( \frac{3}{5} \)