Give reasons for your answers. Let f(x)=|x^3-9 x| a. Does f^'(0) exist? b. Does f^'(3) exist? c.

step 1 For this question, we're getting the function F of X is equal to the absolute value of x q minus

Give reasons for your answers. Let f(x)=|x^3-9 x| a. Does...

Key Concepts

Continuity

Continuity is the property of a function to have no jumps or breaks in its graph. Although a function may be continuous at a point, it can still fail to be differentiable there if there is a sharp turn or cusp, making the examination of continuity an essential but not sufficient step in analyzing differentiability and extrema.

Critical Points and Extrema

Critical points are where the derivative is zero or does not exist, and these points are candidates for local or global extrema. Analyzing these points involves determining the behavior of the function around them to decide whether they correspond to minima, maxima, or neither.

Piecewise Functions

Piecewise functions are defined by different formulas in different parts of their domain. When handling absolute value functions, the function is naturally split into segments depending on the sign of the inner expression, necessitating separate differentiation in each piece and careful consideration at the boundary points.

Differentiability

Differentiability refers to the property of a function having a well-defined tangent at a point, which means that the left-hand derivative and right-hand derivative must exist and be equal at that point. This concept is central when examining functions that may exhibit sharp corners or cusps.

Absolute Value Function

The absolute value function transforms its input into a non-negative value, which often results in a piecewise definition. This is crucial because when the inner function changes sign, the behavior of the derivative can change abruptly, potentially leading to points where the function is not differentiable.

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