Problem 38 - College Algebra

Question

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. $$k(t)=3 t^{\frac{2}{3}}-t$$

Step-by-Step Solution

Answer

The function has a local maximum at \( t = 8 \) and it is increasing on \( t < 8 \) and decreasing on \( t > 8 \).

Step 1: Understanding the Function

The given function is \( k(t) = 3t^{\frac{2}{3}} - t \). It is a combination of a fractional power and a linear term. Our goal is to find the local extrema and the intervals over which it is increasing or decreasing.

Step 2: Finding the Derivative

Calculate the derivative of \( k(t) \) to find the critical points. The derivative \( k'(t) \) is found using the power rule and derivative of a linear term: \[ k'(t) = 2t^{\frac{-1}{3}} - 1. \]

Step 3: Setting the Derivative to Zero

Set \( k'(t) = 0 \) to find critical points: \[ 2t^{\frac{-1}{3}} - 1 = 0. \] Solving for \( t \), we get \( 2t^{\frac{-1}{3}} = 1 \), which simplifies to \( t = 8 \).

Step 4: Determining Increasing/Decreasing Intervals

To determine the intervals on which the function is increasing or decreasing, test intervals around the critical point \( t = 8 \).\- For \( t < 8 \), if \( k'(t) > 0 \), the function is increasing. \- For \( t > 8 \), if \( k'(t) < 0 \), the function is decreasing.

Step 5: Estimating Local Extrema

Substitute \( t = 8 \) back into the original function to find the local extremum. Calculate \( k(8) = 3(8^{\frac{2}{3}}) - 8 \). Simplifying, we get \( k(8) = 4 \). This value represents the local maximum of the function.

Key Concepts

Increasing and Decreasing Intervals

To understand where a function is increasing or decreasing, we use the derivative, denoted as \( k'(t) \). It's important to first find where the derivative is equal to zero or undefined, as these are potential critical points. Depending on the derivative's sign in these intervals, we can determine the behavior of the function.

• If \( k'(t) \) is positive in an interval, the function is increasing in that region.
• If \( k'(t) \) is negative, the function decreases.

Therefore, evaluating the sign of \( k'(t) \) around the critical point \( t = 8 \) helps us to know:
- Between critical points, you test values to see how the sign changes.
This sign change will indicate a shift from increasing to decreasing or vice versa.

Derivative Calculation

Calculating the derivative is key to understanding the function's behavior. For this function, \( k(t) = 3t^{\frac{2}{3}} - t \), we calculate the derivative using basic rules:

• The power rule for \( t^{\frac{2}{3}} \) leads to \( 2t^{\frac{-1}{3}} \).
• The linear term \(-t\) becomes \(-1\).

Thus, the derivative is \( k'(t) = 2t^{\frac{-1}{3}} - 1 \). This derivative tells us how the function is moving at any point \( t \). Solving equations like \( k'(t) = 0 \) helps find critical points, guiding us to understand the curve's shape better.

Critical Points

Critical points are where the derivative \( k'(t) \) is zero or undefined. These points might represent peaks, troughs, or points of inflection. In our example, we solved \( k'(t) = 0 \) and found the critical point at \( t = 8 \).
Why are critical points important? At these points, the function could potentially change from increasing to decreasing, or vice versa, marking a local maximum or minimum. Evaluating the original function \( k(t) \) at these points shows the local extremum. In this case, substituting \( t = 8 \) into the original function yields the value \( k(8) = 4 \), which indicates a local maximum.

Graphing Utility

A graphing utility can visually demonstrate the function's behavior and help verify analytical work. With it, you can plot \( k(t) = 3t^{\frac{2}{3}} - t \) and clearly see where the function rises and falls.

• Use the graph to locate approximate critical points visually.
• Confirm calculated intervals of increase and decrease.
• Ensure estimated local extrema align with those calculated.

Graphing tools are excellent resources for cross-verifying your findings. They provide a picture of the function, reinforcing the trust in the analytical results obtained by hand calculations.