91Ó°ÊÓ

The absolute value function is a crucial mathematical tool that measures the "magnitude" of a number without considering its sign. In simpler terms, it makes any negative quantity positive. In our exercise, the function in question is defined as \( f(x) = |x - 5| \). This particular format shows how the function transforms depending on the value of \( x \) in relation to 5.

• If \( x < 5 \), the expression inside the absolute value, \( x - 5 \), becomes negative, and as a result, \( f(x) = 5 - x \).
• If \( x > 5 \), the expression inside is positive, so \( f(x) = x - 5 \).

This function graphically forms a "V-shape" with a vertex at \( x = 5 \). The sharp tip of this V at the vertex provides an essential characteristic relevant for differentiability analysis.

The left-hand derivative is a mathematical concept that describes the rate of change of a function as you approach a point from the left. Essentially, it's a way of capturing how the function behaves just before reaching the specific point.
To calculate the left-hand derivative of \( f(x) = |x - 5| \) at \( x = 5 \), we use the fact that when \( x < 5 \), \( f(x) = 5 - x \). We compute:\[\lim_{h \to 0^-} \frac{f(5 + h) - f(5)}{h}\]Substituting, we get the expression:\[\lim_{h \to 0^-} \frac{5 - (5 + h)}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1\]This step confirms that the left-hand derivative of the function at \( x = 5 \) is \(-1\). This result is critical because differentiability requires both left and right derivatives to exist and to be equal.

The right-hand derivative operates similarly to the left-hand derivative but assesses the rate of change as you approach a point from the right. It tends to emphasize how the function is behaving immediately after the specific point under consideration.
For the function \( f(x) = |x - 5| \), we need to compute this derivative at \( x = 5 \) by considering \( f(x) = x - 5 \) for \( x > 5 \). Our calculation is:\[\lim_{h \to 0^+} \frac{f(5 + h) - f(5)}{h}\]Plugging in the values gives:\[\lim_{h \to 0^+} \frac{(5 + h) - 5}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1\]This result shows that the right-hand derivative of the function at \( x = 5 \) is \(1\). As with the left-hand side, identifying this derivative is necessary to determine the differentiability of the function.

A sharp corner is a point on a graph where there is a distinct change in the direction of the curve, often resulting in an instantaneous and dramatic shift. In the context of differentiability, sharp corners present a unique challenge.
At \( x = 5 \) in the function \( f(x) = |x - 5| \), we observe such a corner. From our previous findings:

• The left-hand derivative is \(-1\).
• The right-hand derivative is \(1\).

These differing derivative values highlight the presence of a sharp corner where the graph changes direction abruptly at \( x = 5 \). This non-smooth transition means the function is not differentiable at this point, as differentiability necessitates that both derivatives be equal. This exemplifies exactly why sharp corners are obstacles to differentiability.