Problem 45 - A Complete Resource Book in Mathematics for JEE Main

Question

The solution set of $$ \left(\frac{3}{5}\right)^{x}=x-x^{2}-9 \text { is } $$ (A) \(\\{0\\}\) (B) \(\\{1\\}\) (C) \(\phi\) (D) None of these

Step-by-Step Solution

Answer

(C) \(\phi\)

Step 1: Analyze the Equation

We start with the equation: \(\left(\frac{3}{5}\right)^{x} = x - x^{2} - 9\). We need to find the values of \(x\) that satisfy this equation. The left side is an exponential function, and the right side is a quadratic function.

Step 2: Analyze Behavior of Functions

The exponential function \(\left(\frac{3}{5}\right)^{x}\) is always positive for any real \(x\), decreasing towards zero as \(x\) increases. The quadratic function \(x - x^{2} - 9\) is a downward-facing parabola opening downward, with roots (where it crosses the x-axis) and a maximum point we should determine.

Step 3: Find Roots of Quadratic Function

We will find the roots of the quadratic function \(x - x^{2} - 9\) by solving \(x^2 - x + 9 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which simplifies to \(x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 9}}{2}\). The discriminant is negative \((-35)\), indicating there are no real roots.

Step 4: Determine Maximum Value of Quadratic Function

Find the vertex of the parabola \(x - x^{2} - 9\) to determine its maximum value using \(x = \frac{-b}{2a}\). Here, \(a = -1\) and \(b = 1\), so the maximum occurs at \(x = \frac{1}{2}\). Evaluating the function at \(x = \frac{1}{2}\), we get \(\frac{1}{2} - \left(\frac{1}{2}\right)^{2} - 9 = \frac{1}{2} - \frac{1}{4} - 9 = \frac{-35}{4}\), which is negative.

Step 5: Compare Function Values

The maximum value of the quadratic \(x - x^2 - 9\) is negative \((\frac{-35}{4})\), while the exponential function \(\left(\frac{3}{5}\right)^{x}\) is always positive. Therefore, no solution \(x\) can satisfy the equation \(\left(\frac{3}{5}\right)^{x} = x - x^{2} - 9\) because the functions never intersect.

Step 6: Conclusion

Since there are no values of \(x\) where the functions intersect, the solution set of the equation is empty. Therefore, the correct answer is \(\phi\).

Key Concepts

Exponential functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In the given problem, the exponential function is \((\frac{3}{5})^{x}\).
This function is unique in that it changes steadily as \(x\) varies. Specifically, because the base \(\frac{3}{5}\) is less than 1, the function is decreasing.
This means as \(x\) increases, \(\left(\frac{3}{5}\right)^{x}\) gets closer to zero, but it never becomes zero or negative.
This is a key characteristic of exponential functions:

• When the base is between 0 and 1, the function decreases.
• When the base is greater than 1, it increases.

This behavior is important in understanding why an exponential function like \(\left(\frac{3}{5}\right)^{x}\) in the given problem can never intersect with a quadratic function that becomes negative, since exponential functions are always positive.

Quadratic functions

Quadratic functions take the form \(ax^2 + bx + c\), producing a parabola when graphed. In the problem, the quadratic function is \(x - x^2 - 9\).
This creates a downward-opening parabola. A parabola's shape is determined by the coefficient \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

The sum of terms \(x - x^2 - 9\) leads to the parabola having no real roots, as seeing through solving \(x^2 - x + 9 = 0\) and realizing the negative discriminant \(-35\) indicates no intersection with the x-axis.
The maximum point, the vertex, is determined using \(x = \frac{-b}{2a}\), found here at \(x = \frac{1}{2}\).
Evaluating it gives a negative vertex of \(\frac{-35}{4}\).
This means entire parabola lies below \(y = 0\), making it impossible for the parabola to meet with the positive exponential curve.

Solution of equations

Solving equations involves finding values of variables that satisfy two expressions being equal. Here, we try to find \(x\) for which \((\frac{3}{5})^{x} = x - x^2 - 9\).
This requires examining where two different types of functions might intersect. However, we learned earlier the exponential function is always positive, while the quadratic function reaches no positive values.
Explicit steps to solve equations involving different function types include:

• Comparing behaviors: Check if it’s possible for given functions based on their nature to intersect or match each other’s value.
• Finding Roots: Determining intersection points by solving each equation’s root when they are possible.
• Evaluating Expressions: Checking valuation at significant points, such as maximum or minimum for quadratics.

In conclusion, if their inherent properties (such as positivity) contradict, as in the problem, they clearly won't ever share values.
This concept of function intersection is a fundamental perspective in understanding complex equations and can save time by illustrating when solutions do not exist.