Let `f(x) lt 0 AA x in (-oo, 0) and f (x) gt 0 AA x in (0,oo) ` also `f (0)=0,` Again `f'(x) lt 0 AA x in (-oo, -1) and f '(x) gt 0 AA x in (-1,oo)` also `f '(-1)=0` given `lim _(x to -oo) f (x)=0 and lim _(x to oo) f (x)=oo` and function is twice differentiable.
The minimum number of points where `f'(x)` is zero is: (a) 1 (b) 2 (c) 3 (d) 4

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will analyze the given information about the function \( f(x) \) and its derivative \( f'(x) \).

### Step 1: Analyze the behavior of \( f(x) \)
- We know that \( f(x) < 0 \) for \( x \in (-\infty, 0) \) and \( f(x) > 0 \) for \( x \in (0, \infty) \).
- Additionally, \( f(0) = 0 \).
- This indicates that the function crosses the x-axis at \( x = 0 \) and is negative to the left of 0 and positive to the right.

**Hint:** Consider the implications of the function being negative and positive in different intervals.

### Step 2: Analyze the behavior of \( f'(x) \)
- We have \( f'(x) < 0 \) for \( x \in (-\infty, -1) \) and \( f'(x) > 0 \) for \( x \in (-1, \infty) \).
- Also, \( f'(-1) = 0 \), which indicates that \( x = -1 \) is a critical point.

**Hint:** The sign of the derivative tells us about the increasing or decreasing nature of the function.

### Step 3: Determine the minimum number of critical points
- Since \( f'(-1) = 0 \), we have at least one critical point at \( x = -1 \).
- For \( x < -1 \), \( f'(x) < 0 \) indicates that the function is decreasing.
- For \( x > -1 \), \( f'(x) > 0 \) indicates that the function is increasing.

**Hint:** A function can only change from decreasing to increasing at a critical point.

### Step 4: Consider the limits
- We know that \( \lim_{x \to -\infty} f(x) = 0 \) and \( \lim_{x \to \infty} f(x) = \infty \).
- This means as \( x \) approaches negative infinity, \( f(x) \) approaches 0 from below, and as \( x \) approaches positive infinity, \( f(x) \) goes to infinity.

**Hint:** The limits help us understand the end behavior of the function.

### Step 5: Conclusion
- Since \( f'(x) \) is negative for \( x < -1 \) and positive for \( x > -1 \) with only one critical point at \( x = -1 \), there are no other points where \( f'(x) = 0 \).
- Therefore, the minimum number of points where \( f'(x) \) is zero is **1**.

### Final Answer
The minimum number of points where \( f'(x) \) is zero is:
**(a) 1**