Problem 64 - Calculus Early Transcendentals

Question

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. b. Use a graphing utility to find all the solutions to the equation on the given interval. c. Illustrate your answers with an appropriate graph. $$x \ln x-1=0 ;(1, e)$$

Step-by-Step Solution

Answer

Based on the given step-by-step solution, the short answer for this problem is: The function $$f(x) = x \ln x - 1$$ has a solution on the interval (1, e) according to the Intermediate Value Theorem, as it is continuous on this interval, and it crosses the x-axis between these points. By using a graphing utility, we find that the solution is approximately located at x = 1.763.

Step 1: Check if the function is continuous

To apply the Intermediate Value Theorem, we first need to verify if the function $$f(x) = x \ln x - 1$$ is continuous on the interval (1, e). Since the natural logarithm function and the identity function are both continuous, and the product and subtraction of continuous functions are also continuous, we can conclude that the function $$f(x) = x \ln x - 1$$ is continuous on the interval (1, e).

Step 2: Evaluate the function at the endpoints of the interval

Now we need to check if the function takes on values between which we expect it to have a root. Evaluate the function at the interval endpoints: $$f(1) = 1 \ln 1 - 1 = -1$$, and $$f(e) = e \ln e -1 = e-1$$. We have $$f(1) = -1 < 0$$ and $$f(e) = e-1 > 0$$, so the function crosses the x-axis between these two points.

Step 3: Apply the Intermediate Value Theorem

Since the function is continuous on the interval (1, e) and $$f(1) < 0$$, while $$f(e) > 0$$, we can apply the Intermediate Value Theorem. It guarantees that there exists a number c in the interval (1, e) such that $$f(c) = 0$$.

Step 4: Use a graphing utility to find the solutions

To find the exact solution (or solutions) to $$x \ln x - 1 = 0$$ on the given interval (1, e), we can use a graphing utility (such as Desmos, GeoGebra, or a graphing calculator). By graphing the function $$f(x) = x \ln x - 1$$, we can visually identify that there is one root in the interval (1, e), which is approximately at 1.763.

Step 5: Illustrate the answer with a graph

- Plot the function $$f(x) = x \ln x - 1$$ on a graph. - Mark the x-axis with the interval (1, e). - Highlight the point where the function crosses the x-axis in the interval (1, e), which is approximately at (1.763, 0). The graph should confirm that the function has exactly one solution between 1 and e, as shown by the function crossing the x-axis at approximately x = 1.763.