Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=x \sqrt{x+1} $$

Short answer

The critical numbers are x=-1 and x=0. The function is decreasing on (-\( \infty \), -1) and increasing on (-1, 0) and (0, \( \infty \)).

Step-by-step solution

Compute the Derivative

To find the critical numbers, we first need to find the derivative of the function \(f(x)=x \sqrt{x+1}\). This involves applying the product rule, which states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first. We have \(f'(x) = \sqrt{x+1} + \frac{x}{2 \sqrt{x+1}}\). Simplifying this gives us \(f'(x) = \frac{3x + 2\sqrt{x+1}}{2\sqrt{x+1}}\).

Find the Critical Numbers

The critical numbers are the points where the derivative is equal to zero or does not exist. So, we set \(f'(x)\) to be zero and solve the equation. We get that the critical numbers are \(x=-1\) and \(x=0\).

Determine Increasing and Decreasing Intervals

We can now determine where the function is increasing or decreasing. We know that the function is increasing where its derivative is positive, and decreasing where the derivative is negative. So, we test the intervals between the critical numbers (-\( \infty \), -1), (-1, 0), and (0, \( \infty \)) in the derivative. We find that the function is decreasing in (-\( \infty \), -1), increasing in (-1, 0), and increasing in (0, \( \infty \)).

Key concepts

Critical Numbers

In calculus, critical numbers are vital points that help us understand the behavior of **functions**. A critical number of a function is a value of the variable where the derivative of the function is zero or does not exist.
This means that critical numbers often correspond to local maxima, minima, or points of inflection.

To find critical numbers, you follow these steps:

• Take the derivative of the function to understand how it behaves at different points.
• Set the derivative equal to zero and solve for the variable.
• Additionally, identify where the derivative does not exist but the function does.
  

Once you have the critical numbers, these values help in determining specific intervals where the function changes behavior. In our case, with the function \( f(x)=x \sqrt{x+1} \), we found critical numbers at \(x=-1\) and \(x=0\). These points tell us where potential shifts in increasing or decreasing patterns might occur.

Derivative

The concept of the **derivative** is a cornerstone in calculus and is instrumental in understanding how functions change. The derivative of a function at any point measures the rate of change of the function's value with respect to a change in the input value. It's essentially the slope of the tangent line at any given point on the function’s graph.

For the function \(f(x)=x \sqrt{x+1}\), deriving involved using the product rule. The product rule is necessary when the function is a product of two other functions and is defined as follows: if you have two functions, \( u(x) \) and \( v(x) \), their derivative is \( u'(x)v(x) + u(x)v'(x) \).

In our example:

• \(u(x) = x\) and \(v(x) = \sqrt{x+1}\)
• The derivative, \(f'(x)\), then becomes \(\sqrt{x+1} + \frac{x}{2\sqrt{x+1}}\).

After simplifying, we got \(f'(x) = \frac{3x + 2\sqrt{x+1}}{2\sqrt{x+1}}\). This expression tells us about the slope and rate of change across the graph of the function.

Increasing and Decreasing Intervals

Once the critical numbers are identified and the derivative is calculated, we can determine the **increasing and decreasing intervals** of the function. These intervals tell us where the function is 'going up' or 'going down', like a rollercoaster.

To find these intervals:

• Use the derivative of the function.
• Test intervals around the critical numbers to see if the derivative is positive or negative.

If the derivative is positive in an interval, the function is increasing, which means it is headed upward as you move from left to right.
If it’s negative, the function is decreasing.

For our function \(f(x)=x \sqrt{x+1}\), after computing and testing, we found:

• The function decreases on \((-\infty, -1)\).
• It increases on \((-1, 0)\) and \((0, \infty)\).

This information provides insight into the behavior of the function in the vicinity of its critical numbers, painting a clearer picture of the function's graphical nature.