Question

, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ g(x)=|x| ;[-2,2] $$

Short answer

The Mean Value Theorem does not apply because the function is not differentiable at \(x = 0\) on \([-2, 2]\).

Step-by-step solution

Verify Continuity

The first requirement for the Mean Value Theorem (MVT) is that the function must be continuous on the closed interval \([-2, 2]\). \(g(x) = |x|\) is continuous everywhere, hence it is continuous on \([-2, 2]\).

Verify Differentiability

For MVT, the function must be differentiable on the open interval \((-2, 2)\). \(g(x) = |x|\) is not differentiable at \(x = 0\) because the derivative from the left \(g'(-x) = -1\) does not equal the derivative from the right \(g'(x) = 1\). Hence, \(g(x)\) is not differentiable on \((-2, 2)\).

Conclusion for Mean Value Theorem

Since \(g(x)\) is not differentiable on \((-2, 2)\), the Mean Value Theorem does not apply to this function on the given interval \([-2, 2]\).

Sketch the Graph

The graph of \(g(x) = |x|\) can be visualized as a 'V' shape that intersects the origin \((0, 0)\). On the interval \([-2, 2]\), the graph slopes downwards from \((-2, 2)\) to \((0, 0)\) and then upwards from \((0, 0)\) to \((2, 2)\). This visually confirms the non-differentiability at \(x = 0\).

Key concepts

Continuity of Functions

When we talk about the continuity of functions, we're considering whether a function has any breaks, jumps, or holes in its graph on a particular interval. A function is continuous on an interval if, for every point in that interval, you can draw the function without lifting your pencil off the paper.

Mathematically, a function is continuous if the limit of the function as it approaches any point from either direction is equal to the value of the function at that point. For our function, \(g(x) = |x|\), it is continuous everywhere because the absolute value function does not have any breaks or jumps, even through zero. It smoothly transitions values, making it continuous over the closed interval \([-2, 2]\).

This continuity is one of the prerequisites for applying the Mean Value Theorem, a pivotal concept in calculus.

Differentiability

Differentiability is a step further from continuity. If a function is differentiable at a point, it means that the derivative (or slope of the tangent line) exists at that point. If a function is differentiable over an interval, its derivative must exist at every point in that interval.

For the Mean Value Theorem to hold, a function not only needs to be continuous but also differentiable over the open interval. But here's the catch — while all differentiable functions are continuous, not all continuous functions are differentiable.

Take \(g(x) = |x|\) as an example. Though continuous, \(g(x) = |x|\) is not differentiable at \(x = 0\) due to a sharp "corner" in the graph. The derivative from the left is \(-1\), while from the right it is \(+1\), thus not providing a single slope or tangent line at that point.

Piecewise Functions

A piecewise function is a function composed of multiple sub-functions, each of which is applied to a certain interval's subset. This type of function allows different expressions over different sections of its domain. It’s like joining multiple mini-functions to form a comprehensive one.

Our function \(g(x) = |x|\) can be thought of as piecewise because it behaves differently on either side of zero. It can be defined as:

\[g(x) = \begin{cases} -x & \text{for } x < 0 \x & \text{for } x \geq 0\end{cases}\]

This formulation helps visualize why \(g(x) = |x|\) is not differentiable at \(x = 0\). In real-world terms, this means the function has a corner or a cusp at zero where the rule for positive numbers doesn't smoothly transition to the rule for negative numbers.

Absolute Value Functions

Absolute value functions are a fundamental concept in mathematics, particularly in calculus and algebra. These functions measure how far a number is from zero on the number line regardless of direction, always yielding non-negative results.

The absolute value of a number \(x\) is expressed as \(|x|\), meaning:

\[|x| = \begin{cases} x, & \text{if } x \geq 0 \-x, & \text{if } x < 0\end{cases}\]

This characteristic helps these functions form sharp, distinctive V-shaped graphs. Such forms are crucial since they often cause issues with differentiability due to their corners. For the Mean Value Theorem's application, it's crucial to analyze such points to understand the limitations of applying certain calculus principles.

The \(V \) shape in the graph of \(g(x) = |x|\) effectively illustrates its absolute value nature and why it fails the differentiability test at \(x = 0\).