Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{2 x}{x+1}} $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x)=\sqrt{\frac{2 x}{x+1}} \) is \( f'(x)=\frac{1}{(x+1) \sqrt{\frac{2 x}{x+1}}} \). When the derivative is zero, the function is likely at a maximum, minimum or saddle point.

Step 1: Differentiate the Given Function

The function is \( f(x)=\sqrt{\frac{2 x}{x+1}} \). With a small rearrangement we can express it as \( f(x)=(2 x/(x+1))^{1/2} \). Now, we can use the chain rule and power rule to find the derivative. The derivative is \( f'(x)=\frac{1}{2}(2 x/(x+1))^{-1/2} * \frac{d}{dx}(2 x/(x+1)) \). Then, \( f'(x)=\frac{1}{2}(2 x/(x+1))^{-1/2} * (2(x+1) - 2x) / (x+1)^2 = \frac{x+1- x}{(x+1) (2x/(x+1))^{1/2}} \), so \( f'(x)=\frac{1}{(x+1) \sqrt{\frac{2 x}{x+1}}} \).

Step 2: Graph the Function and its Derivative

Both the function \( f(x) \) and its derivative \( f'(x) \) should be graphed in the same viewing window. Note the points where the derivative equals zero or undefined.

Step 3: Analyze the Behavior of the Function when the Derivative is Zero

When the derivative is zero, it implies that the function is at either a maximum, minimum, or saddle point. Looking at the graph, locate the points where the derivative is zero and observe the behavior of the function at these points.