Problem 50 - 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. $$ f(x)=\sqrt{x}\left(2-x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative of the given function \(f(x) = \sqrt{x}(2 - x^2)\) is \(f'(x) = x^{-0.5}(2 - x^2) - 2x\sqrt{x}\). When the derivative of the function is zero, the function is at a local maximum, local minimum, or a saddle point. The behavior of the function can be visualized using its graph and identifying where its slope (derivative) is zero.

Step 1: Symbolic Differentiation

Begin by differentiating the function symbolically. The function \(f(x) = \sqrt{x}(2 - x^2)\) is a product of two functions i.e., \(u(x) = \sqrt{x}\) and \(v(x) = (2 - x^2)\). Hence, the product rule will be used which states \((uv)' = u'v + uv'\). By applying this rule, the derivative becomes \(f'(x) = 0.5x^{-0.5}(2 - x^2) + \sqrt{x}*(-2x)\). Simplifying this will lead to \(f'(x) = x^{-0.5}(2 - x^2) - 2x\sqrt{x}\).

Step 2: Analyzing the derivative

Next, note the behavior of the function where the derivative is zero. This occurs when \(x^{-0.5}(2 - x^2) - 2x\sqrt{x} = 0\). Solving this equation for \(x\) will give the points where the slope of the function is zero. In other words, these points are where the graph of the function is flat.

Step 3: Graph the function and its derivative

Plot the graph for both the function and its derivative in the same viewing window. Based on the plotted graph, we can comment on how the function behaves.

Step 4: Interpretation of Graphs

Based on the graph, one can notice that when the derivative is zero, the function is either at a local maximum, local minimum, or a saddle point (point of inflection). These points can be better understood as the 'hills' and 'valleys' of the graph. They are points where the function changes from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). A saddle point is a point where the function has a zero slope but does not have a local maximum or minimum.