Problem 99 - Calculus and its applications

Question

Find the interval(s) for which \(f^{\prime}(x)\) is positive. Use the derivative to help explain why \(f(x)=x^{5}+x^{3}\) increases for all \(x\) in \((-\infty, \infty)\).

Step-by-Step Solution

Answer

The derivative \( f'(x) = 5x^4 + 3x^2 \) is positive for all \( x \in (-\infty, \infty) \), so \( f(x) \) increases everywhere.

Step 1: Find the Derivative

To determine where the function is increasing, we need the derivative. Start by finding the derivative of \( f(x) = x^5 + x^3 \) using the power rule. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \), so apply it to each term separately. The derivative \( f'(x) \) is \( 5x^4 + 3x^2 \).

Step 2: Analyze the Sign of the Derivative

To determine where \( f'(x) \) is positive, analyze the expression \( 5x^4 + 3x^2 \). Notice that both terms are positive for any real \( x \), because squares of real numbers are non-negative. Therefore, \( 5x^4 + 3x^2 \) is always positive.

Step 3: Identify the Interval

Since \( 5x^4 + 3x^2 \) is positive for all \( x \in (-\infty, \infty) \), the derivative \( f'(x) \) is positive for this entire interval. Thus, \( f(x) \) increases for all real numbers \( x \).

Step 4: Conclusion

Since there are no critical points or intervals where the derivative is zero or negative, \( f(x) = x^5 + x^3 \) increases for all \( x \) in \( (-\infty, \infty) \).

Key Concepts

Derivative

In calculus, a derivative is a measure of how a function changes as its input changes. Think of it as the "rate of change" or the "slope" of the function at any given point. For example, if you have a function that describes the position of a car over time, the derivative of this function will tell you the speed of the car at any particular moment.
To find the derivative of a function, we apply rules such as the power rule, among others. These rules help us understand how the function behaves. For the function given here, \(f(x) = x^5 + x^3\), the derivative \(f'(x) = 5x^4 + 3x^2\) provides valuable information about the nature of the function's increase or decrease across its domain.

Power Rule

One of the most frequently used formulas in calculus is the power rule. It's a handy tool for finding the derivative of functions in the form \( x^n \). According to the power rule, the derivative of \( x^n \) is \( nx^{n-1} \). This rule is simple yet powerful, making the process of finding derivatives straightforward.
When applying the power rule to \(f(x) = x^5 + x^3\), we treat each term separately:

• For \(x^5\), the derivative is \(5x^4\) (since 5 times \(x^{5-1}\) is \(5x^4\)).
• For \(x^3\), the derivative is \(3x^2\) (since 3 times \(x^{3-1}\) is \(3x^2\)).

This gives us a complete derivative of \(5x^4 + 3x^2\). The power rule simplifies the task and ensures accuracy, facilitating a deeper understanding of how the function behaves over its domain.

Increasing Functions

An increasing function is one where, as the input or \(x\) value increases, the output or \(f(x)\) value also increases. This concept is visually observable on a graph as it slopes upwards from left to right. A function is considered increasing on an interval if its derivative is positive for every point within that interval.
For the function \(f(x) = x^5 + x^3\) we discussed, we found its derivative to be \(f'(x) = 5x^4 + 3x^2\). To understand where the function is increasing, look at the sign of the derivative. Here, both \(5x^4\) and \(3x^2\) are never negative for any real number. Both terms become positive for any real value of \(x\), ensuring that \(f'(x)\) is always greater than zero.
Since \(f'(x) > 0\) for all values of \(x\) in \((-\infty, \infty)\), \(f(x) = x^5 + x^3\) is increasing on this entire interval. This tells us that as \(x\) continues to grow, \(f(x)\) keeps getting larger, thus confirming that \(f(x)\) is an increasing function across its whole domain.