Question

Let \[ f(x) = \begin{cases} \frac{7}{2}, & x = 0 \\ \frac{6x + \sin x}{2x + \sin x}, & x \neq 0 \end{cases} \] Then, which of the following statements are correct?

• A. \(f(x)\) has a local maxima at \(x = 0\)
• B. \(f(x)\) has a local minima at \(x = 0\)
• C. Number of maxima in interval \((2\pi, 4\pi)\) is 1
• D. Number of minima in interval \((0, 6\pi)\) is 3

Hint:

For piecewise functions, always check continuity before testing for local maxima or minima.

Answer 1

The Correct Option is C

• At \( x = 0 \), \( \lim f(x) = \frac{7}{3} \neq f(0) = \frac{7}{2} \) ⇒ discontinuity ⇒ no extremum at \( x = 0 \)
• For \( x \neq 0 \), the function is smooth and periodic due to \( \sin x \), and extrema occur periodically.
• Analysis or graphing shows one local maximum in \( (2\pi, 4\pi) \) and 3 local minima in \( (0, 6\pi) \)