https://file.aiquickdraw.com/mathbot/file/1727623689834iaxims86.jpg

The question asks to show that the first derivative \( f'(z) \) and the second derivative \( f''(z) \) exist everywhere, and then to find both \( f'(z) \) and \( f''(z) \) when \( f(z) = z^3 \).

### Step 1: Show that \( f'(z) \) exists and find it
We are given the function \( f(z) = z^3 \). To find \( f'(z) \), we take the derivative of \( f(z) \) with respect to \( z \).

\[
f'(z) = \frac{d}{dz}(z^3) = 3z^2
\]

Thus, \( f'(z) = 3z^2 \), and since \( 3z^2 \) is a polynomial function, the derivative exists everywhere on the complex plane or the real number line.

### Step 2: Show that \( f''(z) \) exists and find it
Next, we find \( f''(z) \) by differentiating \( f'(z) \):

\[
f''(z) = \frac{d}{dz}(3z^2) = 6z
\]

So, \( f''(z) = 6z \), and since \( 6z \) is also a polynomial, the second derivative exists everywhere as well.

### Conclusion:
- \( f'(z) = 3z^2 \)
- \( f''(z) = 6z \)
  
Both the first and second derivatives exist everywhere since they are polynomials.

Would you like more details on any step?

Here are five related questions you might find useful:
1. How do you calculate higher-order derivatives for a given function?
2. What is the geometric interpretation of the first and second derivatives?
3. Can we determine points of inflection using the second derivative for this function?
4. How does the behavior of the function \( f(z) = z^3 \) change based on the values of \( z \)?
5. What role does differentiability play in analyzing polynomial functions?

**Tip:** When working with polynomials, the derivatives are always well-defined and exist everywhere, as polynomials are infinitely differentiable functions.

Show that f'(z) and its derivative f''(z) exist everywhere and find f'(z) and f''(z) when f(z) = z^3.

In this example, we show how to find the first and second derivatives of the function f(z) = z^3. We demonstrate the differentiation process step-by-step and prove that both derivatives exist everywhere on the complex plane.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation