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. $$ f(x)=\frac{\sqrt{x}+1}{x^{2}+1} $$

Short answer

The derivative is \( f'(x) = \frac{2x\sqrt{x}-x^2-\sqrt{x}-1}{(x^2+1)^2} \). The graph of this function and its derivative has been plotted. At the points where the derivative is zero, the original function will either reach a local maximum or minimum or have a point of inflection, which will be evident on the graph.

Step-by-step solution

Finding the Derivative

To find the derivative of the function, apply the quotient rule, which states that if you have a function in the form \( h(x) = \frac{f(x)}{g(x)} \), then the derivative of \( h(x) \) is \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \). Applying this to \( f(x) = \frac{\sqrt{x}+1}{x^{2}+1} \), let the numerator be \( p(x) = \sqrt{x}+1 \) and the denominator be \( q(x) = x^{2}+1 \). Find \( p'(x) \) and \( q'(x) \). Recall that the derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \) and of \( x^n \) is \( nx^{n-1} \). Then use the quotient rule.

Drawing the Graphs

Once the derivative is found, use a graphing utility to plot both the original function and its derivative on the same set of axes. Identify the points where the derivative is zero.

Describing the Behavior

Identify the points where the slope of the tangent lines (the derivative) is zero. Where the derivative is zero, the function has a relative maximum, minimum, or a point of inflection. Analyze the surrounding points to determine which of these applies.