Graph the function f(x)=|x^4-6 x^2|. By looking at the graph and zooming in on points you select

step 1 we have to tell whether this function is a differentiable and continuous at x is equal to 2 so f

Graph the function f(x)=|x^4-6 x^2|. By looking at the graph...

Graph the function $f(x)=\left|x^4-6 x^2\right|$. By looking at the graph and zooming in on points you select, decide at which points the function $f$ has a derivative and at which points it does not. Support your answers with appropriate sketches.

Key Concepts

Graphical Analysis

This concept involves plotting a function to observe its behavior, including continuity, smoothness, and potential points of non-differentiability. Graphical analysis helps identify features like turning points, inflection points, and areas where the slope may change abruptly, which is crucial for understanding the overall behavior of a function.

Absolute Value Functions

Absolute value functions transform an input by taking its non-negative value, which can lead to sharp corners or cusps at points where the expression inside the absolute value changes sign. Understanding this effect is essential when determining differentiability since these kinks typically cause the derivative to be undefined at those transition points.

Differentiability

Differentiability is a measure of whether a function has a well-defined tangent at a given point. A function is differentiable at a point if it is smooth there, meaning there is no abrupt change in direction. In cases where the graph exhibits a corner or cusp, the derivative does not exist at that point.

Piecewise Function Formulation

When dealing with compositions like an absolute value of another function, it is often useful to express the function in a piecewise form. This involves breaking the function into different expressions over intervals defined by where the inside function is non-negative or negative. This formulation aids in analytical examination of the function’s differentiability.

Composite Functions

Composite functions involve the application of one function to the results of another, such as applying the absolute value to a polynomial. Analyzing composite functions requires understanding how the outer function (absolute value) affects the overall behavior, particularly the continuity and differentiability, of the complete function.

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