Minimum integral value of k for which the equation `e^(x)=kx^(2)` has exactly three real distinct solution,

Options

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

Explanation

To find the minimum integral value of \( k \) for which the equation \( e^x = kx^2 \) has exactly three real distinct solutions, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ e^x = kx^2 \]
We can rearrange this to:
\[ f(x) = \frac{e^x}{x^2} = k \]

### Step 2: Analyze the function \( f(x) \)
Define the function:
\[ f(x) = \frac{e^x}{x^2} \]
We need to analyze the behavior of \( f(x) \) to find the values of \( k \) that yield three distinct solutions.

### Step 3: Find the derivative \( f'(x) \)
To find the critical points of \( f(x) \), we compute the derivative:
\[ f'(x) = \frac{(e^x \cdot x^2)' - (e^x \cdot (x^2)')}{(x^2)^2} \]
Using the quotient rule, we get:
\[ f'(x) = \frac{e^x x^2 - e^x \cdot 2x}{x^4} = \frac{e^x (x^2 - 2x)}{x^4} \]
Setting \( f'(x) = 0 \) gives us:
\[ e^x (x^2 - 2x) = 0 \]
Since \( e^x \) is never zero, we solve:
\[ x^2 - 2x = 0 \Rightarrow x(x - 2) = 0 \]
Thus, \( x = 0 \) or \( x = 2 \).

### Step 4: Determine the nature of critical points
We check the second derivative or the sign of \( f'(x) \) around the critical points:
- For \( x < 0 \), \( f'(x) < 0 \) (decreasing).
- At \( x = 0 \), \( f(x) \) approaches infinity.
- For \( 0 < x < 2 \), \( f'(x) > 0 \) (increasing).
- At \( x = 2 \), we check \( f(2) = \frac{e^2}{4} \).
- For \( x > 2 \), \( f'(x) < 0 \) (decreasing).

### Step 5: Find the minimum value of \( k \)
The function \( f(x) \) has a minimum at \( x = 2 \):
\[ f(2) = \frac{e^2}{4} \]
For the equation \( e^x = kx^2 \) to have exactly three distinct solutions, \( k \) must be greater than this minimum value:
\[ k > \frac{e^2}{4} \]
Calculating \( \frac{e^2}{4} \):
Using \( e \approx 2.718 \):
\[ e^2 \approx 7.389 \Rightarrow \frac{e^2}{4} \approx 1.847 \]
Thus, the smallest integer \( k \) satisfying this inequality is:
\[ k \geq 2 \]

### Conclusion
The minimum integral value of \( k \) for which the equation \( e^x = kx^2 \) has exactly three real distinct solutions is:
\[ \boxed{2} \]