Let set of all possible values of `lamda` such that `f (x)= e ^(2x) - (lamda+1) e ^(x) +2x` is monotonically increasing for `AA x in R` is `(-oo, k].` Find the value of k.

Answer

To solve the problem, we need to find the value of \( k \) such that the function

\[ f(x) = e^{2x} - (\lambda + 1)e^x + 2x \]

is monotonically increasing for all \( x \in \mathbb{R} \). A function is monotonically increasing if its derivative is non-negative for all \( x \).

### Step 1: Differentiate the function

First, we differentiate \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(e^{2x}) - \frac{d}{dx}((\lambda + 1)e^x) + \frac{d}{dx}(2x) \]

Using the derivatives of the exponential function and the power rule, we get:

\[ f'(x) = 2e^{2x} - (\lambda + 1)e^x + 2 \]

### Step 2: Set the derivative greater than or equal to zero

For \( f(x) \) to be monotonically increasing, we need:

\[ f'(x) \geq 0 \]

This gives us the inequality:

\[ 2e^{2x} - (\lambda + 1)e^x + 2 \geq 0 \]

### Step 3: Substitute \( y = e^x \)

Let \( y = e^x \). Since \( e^x > 0 \) for all \( x \), we can rewrite the inequality as:

\[ 2y^2 - (\lambda + 1)y + 2 \geq 0 \]

### Step 4: Analyze the quadratic inequality

This is a quadratic inequality in \( y \). For the quadratic \( 2y^2 - (\lambda + 1)y + 2 \) to be non-negative for all \( y > 0 \), the discriminant must be less than or equal to zero. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac = (-(\lambda + 1))^2 - 4 \cdot 2 \cdot 2 \]

Calculating this gives:

\[ D = (\lambda + 1)^2 - 16 \]

### Step 5: Set the discriminant less than or equal to zero

We require:

\[ (\lambda + 1)^2 - 16 \leq 0 \]

This simplifies to:

\[ (\lambda + 1)^2 \leq 16 \]

Taking the square root of both sides, we have:

\[ -\sqrt{16} \leq \lambda + 1 \leq \sqrt{16} \]

This simplifies to:

\[ -4 \leq \lambda + 1 \leq 4 \]

### Step 6: Solve for \( \lambda \)

Subtracting 1 from all parts of the inequality gives:

\[ -5 \leq \lambda \leq 3 \]

### Step 7: Identify the interval

The interval of \( \lambda \) is \( (-5, 3] \). The question states that the set of all possible values of \( \lambda \) is \( (-\infty, k] \). Therefore, we can conclude that:

\[ k = 3 \]

### Final Answer

Thus, the value of \( k \) is:

\[ \boxed{3} \]