Problem 55 - Calculus Single Variable

Question

Let $$f(x)=\left\\{\begin{array}{ll} x^{3} & \text { if } x>0 \\\\-x^{3} & \text { if } x \leq 0 .\end{array}\right.$$ a. Prove that \(f^{\prime}(0)\) exists and equals 0 . b. Prove that \(f^{\prime \prime}(0)\) exists and equals 0 . c. Prove that \(f^{\prime \prime \prime}(0)\) does not exist.

Step-by-Step Solution

Answer

\(f'(0) = 0\), \(f''(0) = 0\), \(f'''(0)\) does not exist.

Step 1: Define Derivative at a Point

To prove that the derivative \(f'(0)\) exists, compute the limit \(\lim_{{h \to 0}} \frac{{f(h) - f(0)}}{h}\). Since \(f(0) = -0^3 = 0\), for the derivative at \(x = 0\), the expression simplifies to \(\lim_{{h \to 0}} \frac{{f(h)}}{h}\).

Step 2: Evaluate Directional Limits for First Derivative

For \(h > 0\), \(f(h) = h^3\), so \(\frac{{f(h)}}{h} = \frac{{h^3}}{h} = h^2\). As \(h\) approaches 0 from the positive side, \(h^2\) approaches 0. For \(h < 0\), \(f(h) = -h^3\), so \(\frac{{f(h)}}{h} = \frac{{-h^3}}{h} = -h^2\). As \(h\) approaches 0 from the negative side, \(-h^2\) approaches 0. Both directional limits are 0.

Step 3: Conclude Existence and Value of First Derivative

Since the directional limits \(\lim_{{h \to 0^+}} \frac{{f(h)}}{h} = 0\) and \(\lim_{{h \to 0^-}} \frac{{f(h)}}{h} = 0\) are equal, \(f'(0)\) exists and equals 0.

Step 4: Define Second Derivative at a Point

To find \(f''(0)\), use the definition of the second derivative: \(f''(0) = \lim_{{h \to 0}} \frac{{f'(h) - f'(0)}}{h}\). Since \(f'(0) = 0\), simplify to \(\lim_{{h \to 0}} \frac{{f'(h)}}{h}\).

Step 5: Compute First Derivative for \(x \neq 0\)

For \(x > 0\), the derivative \(f'(x) = 3x^2\). For \(x < 0\), \(f'(x) = -3x^2\).

Step 6: Evaluate Directional Limits for Second Derivative

For \(h > 0\), \(f'(h) = 3h^2\), so \(\frac{{f'(h)}}{h} = \frac{{3h^2}}{h} = 3h\). As \(h\) approaches 0 from the positive side, \(3h\) approaches 0. For \(h < 0\), \(f'(h) = -3h^2\), so \(\frac{{f'(h)}}{h} = \frac{{-3h^2}}{h} = -3h\). As \(h\) approaches 0 from the negative side, \(-3h\) approaches 0.

Step 7: Conclude Existence and Value of Second Derivative

Since the directional limits \(\lim_{{h \to 0^+}} \frac{{f'(h)}}{h} = 0\) and \(\lim_{{h \to 0^-}} \frac{{f'(h)}}{h} = 0\) are equal, \(f''(0)\) exists and equals 0.

Step 8: Define Third Derivative at a Point

To show \(f'''(0)\) does not exist, compute \(f''(h)\) for \(heq 0\) and examine \(\lim_{{h \to 0}} \frac{{f''(h) - f''(0)}}{h}\). Since \(f''(0) = 0\), focus on \(\lim_{{h \to 0}} \frac{{f''(h)}}{h}\).

Step 9: Compute Second Derivative for \(x \neq 0\)

For \(x > 0\), \(f''(x) = 6x\), and for \(x < 0\), \(f''(x) = -6x\).

Step 10: Evaluate Limits for Third Derivative

For \(h > 0\), \(f''(h) = 6h\), so \(\frac{{f''(h)}}{h} = \frac{{6h}}{h} = 6\). For \(h < 0\), \(f''(h) = -6h\), so \(\frac{{f''(h)}}{h} = \frac{{-6h}}{h} = -6\). The directional limits \(\lim_{{h \to 0^+}} 6 = 6\) and \(\lim_{{h \to 0^-}} -6 = -6\) are not equal.

Step 11: Conclude Non-existence of Third Derivative

Since the directional limits are not equal, \(f'''(0)\) does not exist.

Key Concepts

Understanding Piecewise Functions

A piecewise function is a function that is defined by different expressions depending on the value of the independent variable. In this case, the function \[f(x) = \begin{cases} x^{3} & \text { if } x>0 \-x^{3} & \text { if } x \leq 0 \end{cases}\] is defined in two pieces:

• For values of \(x\) greater than 0, the function is defined as \(x^3\).
• For values of \(x\) that are less than or equal to 0, it is defined as \(-x^3\).

This creates a different behavior on either side of \(x = 0\). It's important to realize that piecewise functions are quite common and their uniqueness lies in how they may behave very differently in separate intervals of their domain. For students, this means learning to evaluate each piece independently before considering the function as a whole.

Limits and Continuity

In calculus, understanding limits is crucial for discussing the continuity and differentiability of a function. A limit is the value that a function (or sequence) "approaches" as the input (or index) approaches some value. When we say \[\lim_{h \to 0} \frac{f(h)}{h}\]we're evaluating how the expression behaves as \(h\) gets infinitely close to 0.
For the given piecewise function, its limit behavior helps determine continuity at points of interest:

• Check the limit from both the positive and negative sides.
• For continuity to hold, the limits from both sides must be equal.

When checking for continuity, particularly at points where a piece of the function changes, as is the case at \(x = 0\), it's essential to confirm that the function's value aligns with the limit's prediction. Since both directional limits at \(x = 0\) return the same value, it suggests continuity. However, differentiability requires further check, especially through derivatives calculation.

Higher-Order Derivatives

Moving beyond first and second derivatives, higher-order derivatives provide insight into how a function's curvature and rate of change develop. The task of assessing higher-order derivatives begins with fully understanding the behavior of the lower order derivatives. For the function,\[ f''(0) = 0 \] existed, indicating that the rate of change of the slope at \(x = 0\) was still constant.
However, further exploration into \[f'''(0)\]revealed contradictions:

• The third derivative tries to narrate the function's jerk, or the rate of change of acceleration.
• For \(x > 0\), the function seems to pull away in one direction, whereas for \(x < 0\), it does the opposite.

Since the limit frequencies do not match, \[\lim_{h \to 0^+} 6 eq \lim_{h \to 0^-} -6\], it confirms diverging behaviors preventing the third derivative from existing. These higher-order derivatives serve as tools to deeply understand and predict a function's future behavior across intervals.