What are the critical points for xtan(x)? The derivative is x(sec^2(x)) + tan(x).

Critical points of a function occur where its derivative is zero or undefined. Let's find the critical points for the function \( f(x) = x \tan(x) \).

Given that the derivative of the function is \( f'(x) = x \sec^2(x) + \tan(x) \), we need to solve the equation \( f'(x) = 0 \) for \( x \).

\[ x \sec^2(x) + \tan(x) = 0 \]

To solve for \( x \), we can manipulate the equation:

\[ x \sec^2(x) = -\tan(x) \]

Now, since \( \sec^2(x) = 1 + \tan^2(x) \), we can rewrite the equation as:

\[ x (1 + \tan^2(x)) = -\tan(x) \]

\[ x + x \tan^2(x) + \tan(x) = 0 \]

This is a transcendental equation, which means it cannot be solved algebraically for exact solutions. However, one obvious solution to this equation is when \( x = 0 \), because \( \tan(0) = 0 \). Therefore, \( x = 0 \) is a critical point.

For other potential critical points, we would need to use numerical methods or graphing to approximate the values of \( x \) where \( x \tan^2(x) + \tan(x) = 0 \) besides \( x = 0 \). This is because both the tangent function and its square have infinite discontinuities at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer, and there could be points between these discontinuities where the equation is satisfied.

Additionally, the critical points occur where \( f'(x) \) is undefined, which is at the points where \( \sec^2(x) \) is undefined. This occurs when \( \cos(x) = 0 \), i.e., at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

In summary, the exact critical point we can find analytically is at \( x = 0 \), and to find other critical points, numerical methods or a graphing calculator would be needed.

Extra: In calculus, a critical point of a function of a real variable is a value at which its derivative is zero or undefined. These points are important because they can indicate local maxima, local minima, or points of inflection in the graph of the function.

The derivative of a function gives us the slope of the function at any given point. When the derivative is positive, the function is increasing; when it's negative, the function is decreasing; and when it is zero, we may have a horizontal tangent line, which could imply a local maximum or minimum. However, not all points where the derivative is zero are maxima or minima; they might also be saddle points, which are not extremes in the graph of the function.

In this case, we are working with the function \( f(x) = x \tan(x) \), which combines a polynomial term and a trigonometric function. The trigonometric function tan(x) is periodic and has vertical asymptotes where it is undefined. The periodicity and asymptotes of tan(x) mean that any investigation into the critical points or behavior of \( f(x) \) must consider these characteristics.