Draw the graph of $f(x) = \sqrt{x}\operatorname{cos}(x)$ for $0 ≤ x ≤ 3$. .

1. Write an equation for $f^\prime$.

   Hint (a):

   Use the product rule.

2. Graph $y = f^\prime(x)$ using your graphing calculator. Then, use this graph to locate the $x$-value of the local maximum of $f$ over $0 ≤ x ≤ 3$.

   Hint (b):

   The location of a local maximum on $f\left(x\right)$ can be found where $f^\prime\left(x\right) = 0$ AND changes from positive values to negative values.

3. Likewise, use a graph of $y = f ^{\prime\prime}(x)$ to locate the point of inflection of $f$ for $0 ≤ x ≤ 3$.

   Hint (c):

   A zero on the second derivative that crosses the $x$-axis (or changes sign) indicates a point of inflection on the original function.