Let `f(x)` be a real valued continuous function on `R` defined as `f(x)=x^2e^(-|x|)` The value of `k` for which the curve `y=k x^2(k >0)` intersect the curve `y=e^(|x|)` at exactly two points, is `e^2` (b) `(e^2)/2` (c) `(e^2)/4\ ` (d) `(e^2)/8`

Options

A. `(e^(2))/4ltkltoo`
B. `(e^(2))/2`
C. `(0ltklt=4)/(e^(2))`
D. `(e^(2))/4`

Explanation

To solve the problem, we need to find the value of \( k \) such that the curves \( y = kx^2 \) and \( y = e^{|x|} \) intersect at exactly two points.

### Step 1: Set the equations equal to each other
We start by setting the two equations equal to each other:
\[ kx^2 = e^{|x|} \]

### Step 2: Analyze the equations
Since \( e^{|x|} \) is an even function, we can analyze the intersections for \( x \geq 0 \) and \( x < 0 \) separately. For \( x \geq 0 \), we have:
\[ kx^2 = e^x \]
For \( x < 0 \), we have:
\[ kx^2 = e^{-x} \]

### Step 3: Find the intersection points for \( x \geq 0 \)
We will focus on the case \( x \geq 0 \) first:
\[ kx^2 - e^x = 0 \]
This is a quadratic equation in terms of \( x \).

### Step 4: Find the derivative
To determine the number of intersection points, we can analyze the function:
\[ g(x) = kx^2 - e^x \]
We find the derivative:
\[ g'(x) = 2kx - e^x \]

### Step 5: Find critical points
Set the derivative to zero to find the critical points:
\[ 2kx = e^x \]
This equation will help us determine the behavior of \( g(x) \).

### Step 6: Analyze the second derivative
To find the nature of the critical points, we can analyze the second derivative:
\[ g''(x) = 2k - e^x \]
This will help us determine if we have a maximum or minimum.

### Step 7: Find the conditions for two intersections
For the curves to intersect at exactly two points, the quadratic \( kx^2 \) must be tangent to \( e^{|x|} \) at one point and intersect at another point. This means that the discriminant of the quadratic equation must be zero for one intersection and positive for another.

### Step 8: Set up the discriminant condition
The discriminant \( D \) of the equation \( kx^2 - e^x = 0 \) must be zero:
\[ D = b^2 - 4ac = 0 \]
This leads to the condition on \( k \).

### Step 9: Solve for \( k \)
We can find the specific value of \( k \) by substituting \( x = 2 \) into the equation:
\[ k(2^2) = e^2 \implies 4k = e^2 \implies k = \frac{e^2}{4} \]

### Final Answer
Thus, the value of \( k \) for which the curves intersect at exactly two points is:
\[ \boxed{\frac{e^2}{4}} \]