Question

If \( f(x) = \begin{cases} \sin\left(\frac{\pi x}{2}\right), & x<1 \\ 3 - 2x, & x \ge 1 \end{cases} \), then \( f(x) \) has:

• A. local minimum at \(x=1\)
• B. local maximum at \(x=1\)
• C. Both local maximum and local minimum at \(x=1\)
• D. None of the above

Hint:

Local maximum if function increases before and decreases after the point.

Answer 1

The Correct Option is B

To determine whether \( f(x) \) has a local minimum or maximum at \( x = 1 \), we need to check the behavior of the function on both sides of \( x = 1 \). The function is defined as:

• \( f(x) = \sin\left(\frac{\pi x}{2}\right) \) for \( x < 1 \)
• \( f(x) = 3 - 2x \) for \( x \ge 1 \)

We'll evaluate the function and its derivative at \( x = 1 \) from both sides.

Step 1: Evaluate the function at \( x = 1 \)

• The value at \( x = 1 \) is given by the function defined for \( x \ge 1 \): \(f(1) = 3 - 2 \times 1 = 1\).

Step 2: Check continuity at \( x = 1 \)

The function should be continuous at \( x = 1 \) for it to have a local extremum. Let's verify:

• Left-hand limit (as \( x \to 1^{-} \)): \(\lim_{x \to 1^{-}} f(x) = \sin\left(\frac{\pi}{2}\right) = 1\).
• Right-hand limit (as \( x \to 1^{+} \)): \(\lim_{x \to 1^{+}} f(x) = 3 - 2 \times 1 = 1\).

Since both limits and the function value at \( x = 1 \) are equal, \( f(x) \) is continuous at \( x = 1 \).

Step 3: Evaluate the derivative at \( x = 1 \)

Next, we look at the derivative on either side of \( x = 1 \):

• For \( x < 1 \), \( f'(x) = \frac{\pi}{2} \cos\left(\frac{\pi x}{2}\right) \).
• Right limit as \( x \to 1^{-} \): \(f'(x) = \frac{\pi}{2} \cos\left(\frac{\pi}{2}\right) = 0\).
• For \( x \ge 1 \), \( f'(x) = -2 \).

Step 4: Identify the extremum

Since the derivative from the left (0) is greater than the derivative from the right (-2), it indicates a decrease as \( x \) crosses 1 from left to right:

• This behavior confirms that \( x = 1 \) is a point of local maximum for the function \( f(x) \).

Thus, the correct answer is that the function has a local maximum at \( x = 1 \).