Find the value of the derivative (if it exists) at each indicated extremum. f(x)=4-|x|

Name: Problem 6, Chapter 4: Calculus Early Transcendental Functions
Uploaded: 2021-07-17T00:32:08-08:00
Duration: 1 min 43 s
Channel: Jacquelyn Trost
Description: Find the value of the derivative (if it exists) at each indicated extremum. f(x)=4-|x|

step 1 For this problem, we've been given the function F of X, which equals 4 minus the absolute value

Find the value of the derivative (if it exists) at each...

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Key Concepts

Absolute Value Function

The absolute value function, denoted by |x|, outputs the non-negative value of x. Its graph is V-shaped, which inherently causes a point of non-differentiability at x = 0 because of the sharp 'corner'. This function is used to express distance from zero and often appears in piecewise definitions.

Differentiability at a Point

Differentiability at a specific point means that the function has a well-defined tangent (or derivative) at that point, requiring that the left-hand derivative and right-hand derivative exist and are equal. A function can be continuous at a point yet not differentiable there if the slopes from the left and the right do not match.

Left-hand and Right-hand Derivatives

The concept of left-hand and right-hand derivatives involves computing the limits of the difference quotient as the variable approaches the point of interest from the left and from the right respectively. For functions involving absolute values, such as 4 - |x|, these one-sided limits can differ, indicating that the derivative does not exist at the point where they are unequal, typically at the vertex of the V-shaped graph.

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