The line `y=x` meets `y=ke^(x),kle0` at

Options

A. no point
B. one point
C. two points
D. None of these

Answer

To find the points of intersection between the line \( y = x \) and the curve \( y = k e^x \) (where \( k \leq 0 \)), we can follow these steps:

### Step 1: Set the equations equal to each other
We start by setting the two equations equal to find the points where they intersect:

\[ x = k e^x \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:

\[ x - k e^x = 0 \]

### Step 3: Analyze the function
Let \( f(x) = x - k e^x \). We need to analyze this function to determine the number of solutions (or intersections) it has.

### Step 4: Find the derivative
To find the critical points, we take the derivative of \( f(x) \):

\[ f'(x) = 1 - k e^x \]

### Step 5: Determine the behavior of the derivative
Since \( k \leq 0 \), \( -k \geq 0 \). Therefore, \( f'(x) \) will always be positive because \( e^x > 0 \) for all \( x \). This means that \( f(x) \) is a strictly increasing function.

### Step 6: Evaluate the limits
Now we evaluate the limits of \( f(x) \) as \( x \) approaches negative and positive infinity:

- As \( x \to -\infty \):
\[ f(x) \to -\infty \quad (\text{since } e^x \to 0) \]

- As \( x \to +\infty \):
\[ f(x) \to +\infty \quad (\text{since } x \to +\infty \text{ and } k e^x \text{ grows slower than } x) \]

### Step 7: Apply the Intermediate Value Theorem
Since \( f(x) \) is continuous and strictly increasing, and it goes from \(-\infty\) to \(+\infty\), by the Intermediate Value Theorem, there is exactly one root for \( f(x) = 0 \).

### Conclusion
Thus, the line \( y = x \) meets the curve \( y = k e^x \) at exactly one point.

### Final Answer
The line \( y = x \) meets \( y = k e^x \) at one point of intersection.

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