Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x \arctan x $$

Short answer

The function \(f(x)=x \arctan x\) has one critical number at \(x = 0\). The function is increasing on the intervals \(-\infty, 0\) and \(0, \infty\). There are no relative extrema.

Step-by-step solution

Compute the derivative of the function

For \(f(x)=x \arctan x\), using the product rule, where the derivative of a product of two functions is first function times the derivative of the second plus the second function times the derivative of the first, we get: \(f'(x) = \arctan x + x \cdot \frac{1} {1+x^{2}}\)

Find Critical Points

A critical number of a function is a number where that function's derivative is zero or undefined. So, to find critical points for either increasing or decreasing trends, set derivative equal to zero: \(\arctan x + x \cdot \frac{1} {1+x^{2}} = 0 \) . Solving this equation, we find \(x = 0\) is the only critical point.

Checking for Intervals of Increase or Decrease and relative extrema

Applying the first derivative test with the critical number, we find that for \(x < 0\), \(f'(x) > 0\), meaning the function is increasing on the interval \(-\infty, 0)\), and for \(x > 0\), \(f'(x) > 0\), meaning the function is increasing on the interval \(0, \infty\). Hence, \(x = 0\) is not a local extremum.

Verify with a graphing utility

On a graphing utility, the function \(f(x) = x*\arctan x\) also visually shows that it is increasing from \(-\infty\) to \(\infty\) and there are no relative extrema.