91Ó°ÊÓ

Derivatives are at the heart of calculus, serving as a tool to determine the rate at which a function changes at any given point. For a given function, the derivative helps reveal important characteristics about its behavior. In the function \(f(x) = \sqrt{|x|}\), we first need to express it piecewise to apply derivatives correctly. This involves considering two cases for the expression based on the value of \(x\):

• When \(x \geq 0\), the function simplifies to \(\sqrt{x}\).
• When \(x < 0\), the function simplifies to \(\sqrt{-x}\).

By differentiating both parts, we get \(\frac{1}{2}x^{-\frac{1}{2}}\) for \(x \geq 0\), signifying that the slope is positive.For \(x < 0\), the derivative is \(-\frac{1}{2}(-x)^{-\frac{1}{2}}\), indicating a negative slope. Analyzing these derivatives helps us understand whether the function is increasing or decreasing over these domains.

Critical points occur where the first derivative of a function is zero or undefined. They are pivotal in determining where a function might have a local maximum, minimum, or saddle point. However, in our function \(f(x) = \sqrt{|x|}\), the derivative is never equal to zero for \(x eq 0\).
It's intrinsic to note that for this particular function, \(x = 0\) is a point of interest. At this point, the derivative isn't zero but the function changes from decreasing to increasing.

• For \(x \geq 0\), the derivative is positive, suggesting the function is increasing.
• For \(x < 0\), the derivative is negative, indicating the function is decreasing.

As a result, there are no traditional critical points that lead us to local maxima or minima in this function. The behavior of the function at \(x=0\) provides continuity, easing the transition in the graph.

Graph sketching is a valuable skill that enables visualization of mathematical functions. For \(f(x) = \sqrt{|x|}\), sketching the graph involves using all derived information. We divide the task based on the domains:

• For \(x \geq 0\), the function behaves like \(\sqrt{x}\) which is an increasing curve starting from the origin.
• For \(x < 0\), the function behaves like \(\sqrt{-x}\) making it a decreasing curve again originating from the origin.

The result of combining these evaluations is a V-shaped graph symmetrical about the y-axis, with a minimum point at the origin. The branches extend into the first and second quadrants, vividly showing the transition from decrease to increase at the y-axis.

Even functions have the characteristic property of symmetry about the y-axis. For a function \(f(x)\), this implies that \(f(x) = f(-x)\). Our discussed function \(f(x) = \sqrt{|x|}\) clearly embodies this property, as changing \(x\) to \(-x\) does not alter the outcome of the function.
This symmetrical property is essential for understanding the graph's shape, as it guarantees a mirrored behavior across the y-axis, contributing to its unique V-shape.

• When sketching, this means that we only need to evaluate one half of the graph and then reflect it over the y-axis to complete it.
• This reflection about the y-axis simplifies analysis and computation in symmetrical functions.

Understanding even functions helps in identifying symmetry, ensuring accuracy in graph sketching and mathematical interpretations.