Find the number of real root (s) of the equation `ae ^(x) =1+ x + (x ^(2))/(2),` where a is positive constant.

Answer

To find the number of real roots of the equation \( ae^x = 1 + x + \frac{x^2}{2} \), where \( a \) is a positive constant, we can follow these steps:

### Step 1: Define the functions
Let:
- \( y_1 = ae^x \)
- \( y_2 = 1 + x + \frac{x^2}{2} \)

### Step 2: Differentiate the functions
Now we differentiate both functions with respect to \( x \):
- The derivative of \( y_1 \):
\[ y_1' = \frac{d}{dx}(ae^x) = ae^x \]
Since \( a > 0 \) and \( e^x > 0 \) for all \( x \), we have \( y_1' > 0 \). Thus, \( y_1 \) is an increasing function.

- The derivative of \( y_2 \):
\[ y_2' = \frac{d}{dx}\left(1 + x + \frac{x^2}{2}\right) = 1 + x \]
The function \( y_2' \) is positive when \( x > -1 \) and negative when \( x < -1 \). Thus, \( y_2 \) is a decreasing function for \( x < -1 \) and an increasing function for \( x > -1 \).

### Step 3: Analyze the behavior of the functions
- As \( x \to -\infty \):
- \( y_1 \to 0 \) (since \( ae^x \to 0 \))
- \( y_2 \to \infty \) (since \( 1 + x + \frac{x^2}{2} \to \infty \))

- As \( x \to \infty \):
- \( y_1 \to \infty \) (since \( ae^x \to \infty \))
- \( y_2 \to \infty \) (since \( 1 + x + \frac{x^2}{2} \to \infty \))

### Step 4: Find the intersection points
- At \( x = -1 \):
- \( y_1 = ae^{-1} \)
- \( y_2 = 1 - 1 + \frac{1}{2} = \frac{1}{2} \)

Since \( y_1 \) is increasing and \( y_2 \) is decreasing, and given that \( y_1 \to 0 \) as \( x \to -\infty \) while \( y_2 \to \infty \), there must be exactly one point where \( y_1 \) and \( y_2 \) intersect.

### Conclusion
Thus, the number of real roots of the equation \( ae^x = 1 + x + \frac{x^2}{2} \) is **1**.

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