Question

For \(k>0\), the set of all values of \(k\) for which \(k e^{x}-x=0\) has two distinct roots is (A) \(\left(0, \frac{1}{e}\right)\) (B) \(\left(\frac{1}{e}, 1\right)\) (C) \(\left(\frac{1}{e}, \infty\right)\) (D) \((0,1)\)

Short answer

The set of all values of k for which the equation has two distinct roots is \(\left(\frac{1}{e}, \infty\right)\).

Step-by-step solution

Determine the condition for distinct roots

For the exponential equation to have two distinct roots, its derivative must be positive for some values of x because then the function is increasing at some interval and intersects the x-axis at two points. To find when the derivative is positive, let's differentiate the function with respect to x.

Differentiate the function with respect to x

Differentiate the function \(k e^{x} - x\) to get \(k e^{x} -1\). We will then set the derivative greater than zero to find the range of values for \(k\).

Solve the inequality for k

Solving \(k e^{x} - 1 > 0\), we find \(k > \frac{1}{e^{x}}\). Since \(e^x\) ranges from 0 to infinity as x increases, the reciprocal \(\frac{1}{e^{x}}\) will range from infinity to 0, hence the minimum value of k when the inequality is valid is \( \frac{1}{e} \).

Conclude the set of all values for k

Since for \(k>\frac{1}{e}\), the derivative is positive and for \(k=\frac{1}{e}\), the function is tangent to the x-axis, we conclude that the set of all values of \(k\) for which the equation has two distinct roots is \( k > \frac{1}{e} \).

Key concepts

Exponential Equations

An exponential equation is one in which a variable appears in the exponent. In the context of the JEE Advanced Mathematics exam, these equations demand a solid understanding of exponent rules and the ability to manipulate expressions for the solution of variables. For instance, in the given exercise, the equation involves the natural exponential function, denoted as \(e^x\).

The equation \(k e^{x}-x=0\) is an example where we're asked to figure out the conditions under which this equation will have two distinct roots. In simple terms, we need to find the range of values of \(k\) that will lead to the equation crossing the x-axis at two different points. It's equivalent to finding the values of \(k\) where the graph of the function intersects the horizontal axis twice, indicating two unique solutions.

Derivatives in Mathematics

In mathematics, a derivative represents the rate of change of a function, similar to how the speedometer on a vehicle indicates the rate of change of position over time. The higher the derivative, the faster the change. In the JEE Advanced context, derivatives play a crucial role in determining the behavior of functions—especially in finding local maxima and minima, points of inflection, and understanding the overall shape of a graph.

For our exponential equation, taking the derivative and analyzing its sign provides insight into whether the function is increasing or decreasing at particular values of \(x\). A positive derivative indicates an increasing function while a negative derivative suggests a decrease. In the given exercise, after differentiating the function to find \(k e^{x} -1\), we examined where this expression is positive. As a result, we can conclude about the function’s intersection points with the x-axis—leading us to identify the range of values for \(k\) that will ensure two distinct roots.

Inequalities

Inequalities are an essential part of algebra and appear extensively in the JEE Advanced Mathematics syllabus. They help in depicting the range of solutions for variables within equations that do not have an exact solution but rather a set of possible solutions that satisfy the given condition.

In the process of solving the differential inequality from the exercise, \(k e^{x} - 1 > 0\), we are essentially determining the conditions under which an inequality holds true. Solving this inequality is a two-step process involving isolating the variable \(k\) to one side and then interpreting the result in context to the full range of the function \(e^{x}\). The solution given explains how the value of \(k\) must be greater than the reciprocal of \(e^{x}\) for the inequality to be valid. This analysis leads us to understand that for \(k > \frac{1}{e}\), the derivative of the function is positive—meaning the function is increasing—enabling us to conclude the necessary condition for the original function to have two distinct roots.