Question

Let \( f(t) \) be a real-valued function whose second derivative is positive for \( -\inftyLt;tLt;\infty \). Which of the following statements is/are always true?

• A. \( f(t) \text{ has at least one local minimum.} \)
• B. \( f(t) \text{ cannot have two distinct local minima.} \)
• C. \( f(t) \text{ has at least one local maximum.} \)
• D. \( \text{The minimum value of } f(t) \text{ cannot be negative.} \)

Hint:

For functions with \( f''(t)>0 \): Critical points are always local minima. No local maxima exist. The function is concave up everywhere.

Answer 1

The Correct Option is B

Step 1: Analyzing the second derivative condition. The second derivative \( f''(t)>0 \) implies that the function \( f(t) \) is concave up for all \( t \). This means: - The curve is bent upwards everywhere. - Any critical point of \( f(t) \) (where \( f'(t) = 0 \)) is a local minimum. Step 2: Checking each option: From Option (1): This is a wrong statement. Since \( f(t) \) is concave up everywhere, it must have at least one local minimum at a critical point. From Option (2): This is a correct statement. The function \( f(t) \) can have multiple distinct local minima depending on its behavior. For example, a function can have two or more minima in different intervals. From Option (3): This is a wrong statement. Since \( f''(t)>0 \), there can be no local maxima because the curve does not bend downward at any point. From Option (4): This is a wrong statement. The minimum value of \( f(t) \) depends on its definition and can be negative. For instance, the function \( f(t) = t^2 - 10 \) has a negative minimum value.