The function y has a relative <strong>MAXIMA</strong> at x=0 for
Consider the following for the next two (02) items that follow : A differentiable function f(x) has a local <strong>MAXIMUM</strong> at x=0. Let y=2f(x)+ax-b.
- A. a \gt 0, b=0
- B. for all b and a=0 ✓
- C. for all b \gt 0 only
- D. for all a and b=0
Correct Answer: B. for all b and a=0
Explanation
Calculate the first derivative of y: y' = 2f'(x) + a. For y to have a relative maximum at x=0, we must have y'(0) = 0. Since f'(0) = 0, we get y'(0) = 2(0) + a = a \implies a = 0. The second derivative is y''(0) = 2f''(0) \lt 0, confirming it's a maximum. This condition holds true regardless of the value of b.
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