Question

Use the definitions of increasing and decreasing functions to prove that \(f(x)=x^{3}\) is increasing on \((-\infty, \infty)\).

Short answer

The function \(f(x)=x^{3}\) is increasing on \((-\infty, \infty)\), because its derivative \(3x^2\) is positive for all \(x\).

Step-by-step solution

Calculating the Derivative

First, calculate the derivative of the function \(f(x) = x^3\). The derivative \(f'(x)\) is given by \(3x^2\).

Analyzing the Sign of the Derivative

Next, analyze the sign of the derivative. For all \(x\) in the domain \(-\infty < x < \infty\), \(3x^2\) is positive because the square of any real number is positive.

Making Conclusions

Since the derivative \(f'(x) = 3x^2\) is defined and positive for all \(x\), it can be concluded that the function \(f(x) = x^3\) is increasing over the interval \((- \infty, \infty)\).