Find the range of values of `lambda` for which the point `(lambda,-1)` is exterior to both the parabolas `y^2=|x|dot`

Explanation

To find the range of values of \( \lambda \) for which the point \( (\lambda, -1) \) is exterior to both parabolas given by the equations \( y^2 = -x \) and \( y^2 = x \), we will follow these steps:

### Step 1: Understand the Parabolas
The first parabola \( y^2 = -x \) opens to the left, and the second parabola \( y^2 = x \) opens to the right.

### Step 2: Substitute the Point into the Parabola Equations
We need to determine the conditions under which the point \( (\lambda, -1) \) lies outside both parabolas.

1. For the first parabola \( y^2 = -x \):
\[ (-1)^2 = -\lambda \implies 1 = -\lambda \implies \lambda < -1 \]

2. For the second parabola \( y^2 = x \):
\[ (-1)^2 = \lambda \implies 1 = \lambda \implies \lambda > 1 \]

### Step 3: Combine the Conditions
From the inequalities derived from both parabolas:
- From the first parabola, we have \( \lambda < -1 \).
- From the second parabola, we have \( \lambda > 1 \).

### Step 4: Determine the Range of \( \lambda \)
The conditions \( \lambda < -1 \) and \( \lambda > 1 \) cannot be satisfied simultaneously. Therefore, there is no value of \( \lambda \) for which the point \( (\lambda, -1) \) is exterior to both parabolas.

### Conclusion
The range of values of \( \lambda \) for which the point \( (\lambda, -1) \) lies exterior to both parabolas is:
\[ \text{No values of } \lambda \text{ satisfy the condition.} \]