Question

If \(f(x)=|x|^{3}\), compute \(f^{\prime}(x), f^{\prime \prime}(x)\) for all real \(x\), and show that \(f^{(3)}(0)\) does not exist.

Short answer

Answer: No, the third derivative does not exist at \(x = 0\).

Step-by-step solution

Rewrite the function

We can rewrite the function as: \(f(x) = x^3\) if \(x \geq 0\) \(f(x) = -x^3\) if \(x < 0\)

Compute the first derivative

Using these rewritten expressions, we can differentiate with respect to \(x\): If \(x \ge 0\), \(f'(x) = \frac{d}{dx} x^3 = 3x^2\). If \(x < 0\), \(f'(x) = \frac{d}{dx} (-x^3) = -3x^2\). So \(f'(x) = 3x^2\) if \(x \ge 0\), and \(f'(x) = -3x^2\) if \(x < 0\).

Compute the second derivative

Now we can differentiate again with respect to \(x\): If \(x \ge 0\), \(f''(x) = \frac{d}{dx} 3x^2 = 6x\). If \(x < 0\), \(f''(x) = \frac{d}{dx} (-3x^2) = 6x\). So, \(f''(x) = 6x\) for all real \(x\).

Check the existence of the third derivative at x = 0

To find the third derivative, we need to compute \(\frac{d}{dx} (6x) = 6\) for all real \(x\), except at \(x=0\). We need to check the limit as \(x\) approaches \(0\) from both sides: From the left: \(\lim_{x \to 0^-} \frac{f'(x) - f'(0)}{x - 0} = \lim_{x \to 0^-} \frac{-3x^2}{x} = \lim_{x \to 0^-} -3x = 0\) From the right: \(\lim_{x \to 0^+} \frac{f'(x) - f'(0)}{x - 0} = \lim_{x \to 0^+} \frac{3x^2}{x} = \lim_{x \to 0^+} 3x = 0\) Since the left and right limits are different, the third derivative does not exist at x = 0. So, we can conclude that \(f^{(3)}(0)\) does not exist.

Key concepts

Understanding Derivatives

The concept of derivatives is pivotal to mathematical analysis and various applications in physics, engineering, and economics. A derivative represents the rate at which a function is changing at any given point. In simpler terms, if you consider a function to signify the position of a moving car over time, its derivative would indicate the velocity of the car - how fast its position changes over time.

When computing derivatives, one often encounters piecewise functions, like the absolute value function raised to the third power, which in essence is a function defined by two sub-functions based on the domain of the input value. For instance, with the function given in the exercise, we see that the nature of the function alters at the point where x equals zero, leading us to approach the derivative separately for positive and negative values of x. This results in two different expressions for the derivative, based on whether x is positive or negative.

By finding the first derivative using the known rules of differentiation, we can then proceed to find the second derivative, which tells us about the acceleration of the car, that is, how quickly the velocity is changing. This second step holds crucial significance in understanding the behavior of the function at all points.

Piecewise Functions and Continuity

Piecewise functions, like the one in our exercise, are defined by different expressions depending on the input values. They can often describe real-world situations that have different rules or conditions.

For seamless analysis, we prefer these functions to be continuous and differentiable. However, they can sometimes present points of discontinuity or where the derivative does not exist. This typically happens at the boundaries between the pieces of the function. In the context of this exercise, it is crucial to carefully examine these boundaries to understand the function’s overall behavior and properties.

When we consider the first and second derivatives, we observe consistent behavior across the negative and positive intervals separately. This step-wise approach allows us to individuate the function's characteristics and the intervals in which they apply, simplifying the computation and understanding of the derivatives.

Grasping the Concept of Limits

Limits in calculus allow us to study the behavior of functions as we approach a particular point. They are foundational for defining both derivatives and integrals. For instance, a limit can describe what happens to a function as x approaches a certain value, without necessarily ever reaching that value. This can reveal the existence (or non-existence) of a function's derivative at a point.

When checking for the third derivative in the exercise, we must consider limits to find the behavior as x approaches zero from both sides. If the limits do not match, this indicates a discontinuity in the derivative's behavior, thus leading to the third derivative being non-existent at that point.

It is important to remember that while a function may be smooth and have derivatives of all orders in most of its domain, there might be specific points, usually at the juncture of piecewise intervals, where higher-order derivatives fail to exist. This highlights the intricate balance between continuity, piecewise-defined functions, and their derivatives.