The range of values of ‘a' such that `(1/2)^|x|= x² - a` is satisfied for maximum number of values of 'x'

Options

A. `(-infty,-1)`
B. `(-infty,infty)`
C. (-1,1)
D. `(-1,infty)`

Answer

To find the range of values of 'a' such that the equation \((1/2)^{|x|} = x^2 - a\) is satisfied for the maximum number of values of \(x\), we will analyze the two sides of the equation step by step.

### Step 1: Rewrite the equation
The given equation is:
\[ (1/2)^{|x|} = x^2 - a \]
We can rewrite this as:
\[ \frac{1}{2^{|x|}} = x^2 - a \]

### Step 2: Analyze the left-hand side
The left-hand side, \((1/2)^{|x|}\), is a decreasing function of \(|x|\). As \(|x|\) increases, \((1/2)^{|x|}\) approaches 0. Specifically:
- For \(x = 0\), \((1/2)^{|0|} = 1\).
- As \(|x|\) increases, \((1/2)^{|x|}\) decreases towards 0.

### Step 3: Analyze the right-hand side
The right-hand side, \(x^2 - a\), is a quadratic function that opens upwards. The vertex of this parabola is at \(x = 0\) and the minimum value is \(-a\). As \(x\) moves away from 0, \(x^2 - a\) increases.

### Step 4: Determine intersection points
To find the maximum number of solutions to the equation, we need to consider the intersection points of the two graphs:
1. If \(-a < 1\), then the minimum value of \(x^2 - a\) is less than 1. This means the parabola can intersect with the decreasing function \((1/2)^{|x|}\) at two points.
2. If \(-a = 1\), the parabola touches the line at one point (tangential intersection).
3. If \(-a > 1\), the parabola does not intersect the line, resulting in no solutions.

### Step 5: Establish the range for 'a'
From the analysis:
- For maximum intersections (two solutions), we need \(-a < 1\) which simplifies to:
\[ a > -1 \]
- For one solution, we have \(-a = 1\) which gives \(a = -1\).
- For no solutions, \(-a > 1\) which gives \(a < -1\).

Thus, the range of values of \(a\) for which the equation has the maximum number of solutions (two solutions) is:
\[ a > -1 \]

### Final Answer
The range of values of \(a\) such that \((1/2)^{|x|} = x^2 - a\) is satisfied for the maximum number of values of \(x\) is:
\[ a \in (-1, \infty) \]