Problem 62 - 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. $$-x^{5}-4 x^{2}+2 \sqrt{x}+5=0 ;(0,3)$$

Step-by-Step Solution

Answer

If yes, find the solution and provide an appropriate graph to illustrate the solution. Answer: Yes, the equation \(-x^{5}-4 x^{2}+2 \sqrt{x}+5 = 0\) has a solution in the interval \((0,3)\). Using the Intermediate Value Theorem and a graphing utility, we find that the approximate solution is \(x \approx 1.243\). A graph of the function in the given interval illustrates the intersection point with the x-axis, confirming the found solution.

Step 1: Define the function

Let's define the function $$f(x) = -x^{5}-4 x^{2}+2 \sqrt{x}+5$$ and the given interval \((0, 3)\).

Step 2: Prove the continuity of the function

In order to use the IVT, we need to prove that the function is continuous in the given interval. The function \(f(x)\) is a polynomial combined with a square-root function, both of which are continuous everywhere in their domains. Since the domain of the square-root function is \([0, +\infty)\), the function \(f(x)\) is continuous on the interval \((0, 3)\).

Step 3: Evaluate the function at the endpoints of the interval

We will now evaluate the function at the endpoints of the interval: $$f(0) = -0^5 - 4 \cdot 0^2 + 2 \sqrt{0} + 5 = 5$$ $$f(3) = -3^5 - 4 \cdot 3^2 + 2 \sqrt{3} + 5 \approx -227.536$$ Since \(f(0) > 0 > f(3)\), there must be a value \(c\) in the interval \((0, 3)\) such that the equation \(f(x) = 0\) holds, according to the IVT.

Step 4: Use a graphing utility to find the solution

Using a graphing utility or software like Desmos, GeoGebra or WolframAlpha, plot the function \(f(x)\) on the interval \((0, 3)\), and observe where it intersects the x-axis: Approximate solutions: $$x \approx 1.243$$

Step 5: Illustrate the answer with an appropriate graph

Create a graph of the function \(f(x)\) on the interval \((0, 3)\). Show the intersection point with the x-axis to illustrate the found solution. To summarize, we used the Intermediate Value Theorem to show that the given equation has a solution in the interval \((0, 3)\). We then used a graphing utility to find the approximate solution, and we provided a graph to illustrate the found solution.