Problem 13 - A Textbook of Differential Calculus

Question

The range of values of ' \(a\) ' such that \(\left(\frac{1}{2}\right)^{|x|}=x^{2}-a\) is satisfied for maximum number of values of ' \(x\) ' (a) \((-\infty,-1)\) (b) \((-\infty, \infty)\) (c) \((-1,1)\) (d) \((-1, \infty)\)

Step-by-Step Solution

Answer

The correct range is \((-\infty, \infty)\); in this range, the equation has multiple solutions for \(x\).

Step 1: Understanding the Equation

Given the equation \(\left(\frac{1}{2}\right)^{|x|} = x^2 - a\), we need to determine for which values of \(a\) the maximum number of solutions for \(x\) exists. This requires examining how the values of both sides of the equation behave as functions of \(x\).

Step 2: Analyzing the Left Side of the Equation

Observe that \(\left(\frac{1}{2}\right)^{|x|}\) is a function that decreases exponentially. It reaches its maximum value of 1 when \(|x| = 0\), and as \(|x|\) increases, the value approaches 0. Note that this term is symmetric about \(x = 0\).

Step 3: Analyzing the Right Side of the Equation

The right side, \(x^2 - a\), is a parabola shifted downward by \(a\) units. For every specific \(a\), this parabola will open upwards with the vertex at \((0, -a)\).

Step 4: Finding the Intersection Points

To find maximum intersection points, analyze when the exponential decay (left side) and the upward-opening parabola (right side) intersect. Since they both meet at positive \(x^2\) values, \(a\) must be such that the parabola intersects the curve as it decays from 1 to 0. At maximum, the parabola \(x^2 - a\) intersects with \(1 = \left(\frac{1}{2}\right)^{0}\) at \(x = 0\), ensuring \(-a = 0\) or \(a = 0\).

Step 5: Check if More Intersections Exist

Check around \(a = 0\) for potential intersections at positive and negative \(x\). The solution is maximized when the vertex of the parabola is above the exponential curve's initial value but allows other points to intersect as \(x\) increases positively, particularly between \(-1 < x < 1\) due to symmetry.

Key Concepts

Range of Values

The concept of "range of values" refers to the set of possible outputs that satisfy a particular condition or equation. It is essentially the domain of a dependent variable, given the domain of the independent variables and the nature of the function. In the given problem, the equation is \[\left(\frac{1}{2}\right)^{|x|} = x^2 - a\]We want to identify the values of \(a\) such that this equation has the maximum number of solutions for \(x\). One way to approach this is to look at how the two sides of the equation behave:

• The left-hand side, \(\left(\frac{1}{2}\right)^{|x|}\), is an exponential function that decreases as \(|x|\) increases.
• The right-hand side, \(x^2 - a\), is a parabolic function that shifts vertically depending on \(a\).

To maximize intersections, \(a\) must be chosen such that the parabola overlaps significantly with the decay of the exponential. Specifically, checking the range \(x \in (-1,1)\) helps to capture multiple intersection points due to symmetry about the \(y\)-axis.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In our exercise, the function \(\left(\frac{1}{2}\right)^{|x|}\) is exponential because it takes the absolute value of \(x\) as an exponent. Key characteristics of exponential functions include:

• They grow or decay rapidly based on whether the base is greater than or less than 1.
• In this context, \(\left(\frac{1}{2}\right)^{|x|}\) decays as \(|x|\) increases because the base is between 0 and 1.
• The function is symmetric around the \(y\)-axis due to the absolute value of \(x\).

For \(|x| = 0\), the function achieves its maximum value of 1, indicating a critical point of interest where the exponential intersects with the parabola. Understanding how this function behaves is vital for determining the equation's solutions.

Parabola

A parabola is a curve described by a quadratic function, typically expressed in the form \(y = ax^2 + bx + c\). In this exercise, the function is given by \[y = x^2 - a\]This specific parabola:

• Opens upwards, as the coefficient of \(x^2\) is positive.
• Has its vertex at the point \((0, -a)\), which indicates the lowest point of the parabola when \(a\) decreases the function vertically.
• The "axis of symmetry" is the y-axis, due to the lack of an \(x\)-term shifting it horizontally.

As \(a\) changes, the whole parabola moves up or down, affecting whether and where it will intersect with \(\left(\frac{1}{2}\right)^{|x|}\). For maximum intersections, positioning the vertex to have the parabolic curve intersect the decay of the exponential function is crucial.

Intersection Points

Intersection points are locations where two graphs meet, denoting solutions where both functions have the same value for a given \(x\). In the context of our exercise, these are the solutions to \[\left(\frac{1}{2}\right)^{|x|} = x^2 - a\]Analyzing intersections involves:

• Determining where the exponential decay meets the parabolic growth.
• Understanding there's symmetry which means intersections at positive \(x\) values typically have counterparts at negative \(x\) values.
• Recognizing the maximal number of intersection points occurs when \(a = 0\), aligning the vertex of the parabola with the peak of the exponential, thus ensuring intersections in the range \(-1 < x < 1\).

Optimizing for these intersections provides the set of \(a\) values which make the equation solvable for the most \(x\) values, particularly emphasized when considering the symmetry and nature of both functions involved.