If `f(x)= int_(x^2)^(x^(2) +4) e^(-t^(2) ) dt`, then the function `f(x)` increases in

Options

A. `(-oo,0)`
B. `(0,oo)`
C. `(-1,2)`
D. `(-2,oo)`

Answer

To determine the interval in which the function \( f(x) = \int_{x^2}^{x^2 + 4} e^{-t^2} \, dt \) is increasing, we need to follow these steps:

### Step 1: Differentiate \( f(x) \)
Using the Leibniz rule for differentiating integrals, we can differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} \left( \int_{x^2}^{x^2 + 4} e^{-t^2} \, dt \right) \]

According to the Leibniz rule, we differentiate the upper limit and lower limit:

\[ f'(x) = e^{-(x^2 + 4)^2} \cdot \frac{d}{dx}(x^2 + 4) - e^{-x^4} \cdot \frac{d}{dx}(x^2) \]

Calculating the derivatives of the limits:

- The derivative of the upper limit \( x^2 + 4 \) is \( 2x \).
- The derivative of the lower limit \( x^2 \) is \( 2x \).

Thus, we have:

\[ f'(x) = e^{-(x^2 + 4)^2} \cdot 2x - e^{-x^4} \cdot 2x \]

### Step 2: Factor out common terms
We can factor out \( 2x \):

\[ f'(x) = 2x \left( e^{-(x^2 + 4)^2} - e^{-x^4} \right) \]

### Step 3: Determine when \( f'(x) > 0 \)
For \( f(x) \) to be increasing, we need \( f'(x) > 0 \):

\[ 2x \left( e^{-(x^2 + 4)^2} - e^{-x^4} \right) > 0 \]

This inequality holds when both factors are positive or both are negative.

1. **Case 1: \( 2x > 0 \)** implies \( x > 0 \).
2. **Case 2: \( e^{-(x^2 + 4)^2} > e^{-x^4} \)** simplifies to:

\[ -(x^2 + 4)^2 > -x^4 \quad \Rightarrow \quad (x^2 + 4)^2 < x^4 \]

This inequality does not hold for any real \( x \) since \( (x^2 + 4)^2 \) is always greater than \( x^4 \) for all \( x \).

### Step 4: Analyze the critical point
The only critical point we found is \( x = 0 \). We can check the sign of \( f'(x) \) around this point:

- For \( x < 0 \), \( 2x < 0 \) and \( e^{-(x^2 + 4)^2} - e^{-x^4} \) is positive (since \( e^{-(x^2 + 4)^2} \) is always less than \( e^{-x^4} \)).
- For \( x > 0 \), \( 2x > 0 \) and \( e^{-(x^2 + 4)^2} - e^{-x^4} < 0 \).

### Conclusion
Thus, \( f'(x) > 0 \) only when \( x < 0 \). Therefore, the function \( f(x) \) is increasing in the interval:

\[ (-\infty, 0) \]