Problem 106

Question

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

Step-by-Step Solution

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Answer

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

1Step 1: Calculating the Derivative

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

2Step 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.

3Step 3: 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 $(- \infty, \infty)$.

Problem 105

Problem 107

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