For the function `y=f(x)=(x^(2)+bx+c)e^(x)`, which of the following holds?

Options

A. If `f(x)gt0` for all real `xcancelrArrf'(x)gt0`
B. If `f(x)gt0` for all real `xrArrf'(x)gt0`
C. If `f'(x)gt0` for all real `xrArrf(x)gt0`
D. If `f'(x)gt0` for all real `xcancelrArrf(x)gt0`

Answer

To solve the problem, we need to analyze the function \( y = f(x) = (x^2 + bx + c)e^x \) and determine the validity of the given options regarding the relationship between \( f(x) \) and its derivative \( f'(x) \).

### Step 1: Identify the function and its components
The function is given as:
\[ f(x) = (x^2 + bx + c)e^x \]
Here, \( e^x \) is always positive for all real \( x \).

### Step 2: Differentiate the function using the product rule
To find \( f'(x) \), we will apply the product rule, which states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then:
\[ (uv)' = u'v + uv' \]
Let \( u(x) = x^2 + bx + c \) and \( v(x) = e^x \).

Calculating the derivatives:
- \( u'(x) = 2x + b \)
- \( v'(x) = e^x \)

Now applying the product rule:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
Substituting the values:
\[ f'(x) = (2x + b)e^x + (x^2 + bx + c)e^x \]
Factoring out \( e^x \):
\[ f'(x) = e^x \left( (2x + b) + (x^2 + bx + c) \right) \]
Simplifying further:
\[ f'(x) = e^x \left( x^2 + (b + 2)x + (b + c) \right) \]

### Step 3: Analyze the conditions for positivity
Since \( e^x \) is always positive, the sign of \( f'(x) \) depends on the quadratic expression:
\[ g(x) = x^2 + (b + 2)x + (b + c) \]

### Step 4: Determine conditions for \( g(x) \)
To analyze when \( g(x) \) is positive, we need to consider the discriminant:
\[ D = (b + 2)^2 - 4 \cdot 1 \cdot (b + c) \]
For \( g(x) \) to be positive for all \( x \), we need:
1. \( D < 0 \) (no real roots, hence always positive)
2. The leading coefficient (which is 1) is positive.

### Step 5: Evaluate the options
1. **Option 1**: If \( f(x) > 0 \) for all \( x \), it does not imply \( f'(x) > 0 \).
- This is **True**. If \( f(x) \) is positive but has a local maximum or minimum, \( f'(x) \) can be zero or negative.

2. **Option 2**: If \( f(x) > 0 \), then \( f'(x) > 0 \).
- This is **False**. As shown in the analysis, \( f(x) > 0 \) does not guarantee \( f'(x) > 0 \).

3. **Option 3**: If \( f'(x) > 0 \), then \( f(x) > 0 \).
- This is **True**. If the derivative is positive, the function is increasing, and since it starts from a positive value, it remains positive.

4. **Option 4**: If \( f'(x) > 0 \), it does not imply \( f(x) > 0 \).
- This is **False**. If \( f'(x) > 0 \), it indicates that \( f(x) \) is increasing, and if it started positive, it will remain positive.

### Conclusion
The correct options are:
- Option 1: True
- Option 3: True