The First Derivative Test is a helpful tool in math. It helps us understand how functions work, especially when we're looking for their highest or lowest points, known as maxima and minima. This test is important for solving problems where we want to find the best solution, like in economics, engineering, and science. To use the First Derivative Test well, we need to know the steps, the ideas behind it, and how to read the results.

First, let’s understand what a critical point is. A critical point is where a function’s slope, or derivative, is zero or doesn’t exist. These points are important because they can be where the highest or lowest values occur. For a function called ( f(x) ), we find the derivative, ( f'(x) ). The critical points happen when ( f'(x) = 0 ) or when the derivative cannot be calculated.

Steps to Use the First Derivative Test:

1. Differentiate the Function: Start with a function ( f(x) ) that can be differentiated. Find its first derivative ( f'(x) ).

2. Identify Critical Points: Solve ( f'(x) = 0 ) to find the critical points. Also, see where ( f'(x) ) is undefined because those points are also important.

3. Test Intervals Around Critical Points: Choose test points in the intervals that the critical points create. This means picking points on both sides of each critical point.

4. Evaluate the Sign of the Derivative: Check whether ( f'(x) ) is positive or negative for each test point. If ( f'(x) ) is positive, the function is going up. If ( f'(x) ) is negative, the function is going down.

5. Determine Maximum and Minimum Values: Look at the changes in ( f'(x) ):

   • If it goes from positive to negative at a critical point, that point is a local maximum.
   • If it goes from negative to positive, it’s a local minimum.
   • If there's no change, then the critical point isn’t a max or min.

Example:

Let’s use the First Derivative Test with the function ( f(x) = -2x^2 + 4x + 1 ).

1. Differentiate the Function: Find the derivative: ( f'(x) = -4x + 4 ).

2. Identify Critical Points: Set the derivative to zero: ( -4x + 4 = 0 ) leads to ( x = 1 ). There are no places where the derivative is undefined, so the only critical point is ( x = 1 ).

3. Test Intervals: The critical point divides the number line into two parts: ( (-\infty, 1) ) and ( (1, +\infty) ). Choose test points, like ( x = 0 ) for the first part and ( x = 2 ) for the second.

4. Evaluate the Sign of the Derivative:

   • For ( x = 0 ): ( f'(0) = -4(0) + 4 = 4 ) (positive, so the function is increasing).
   • For ( x = 2 ): ( f'(2) = -4(2) + 4 = -4 ) (negative, so the function is decreasing).

5. Determine Maximum and Minimum Values: Since ( f'(x) ) changes from positive to negative at ( x = 1 ), we know ( f(x) ) has a local maximum there. To find the value at this point, plug it back into the function: ( f(1) = -2(1)^2 + 4(1) + 1 = 3 ). So, the local maximum is ( 3 ) at ( x = 1 ).

Sometimes, there are more factors to think about, like limits or endpoints. If our function is defined between two points, we also have to check those points to find any max or min we might have missed. For example, if our function were limited to the interval ([0, 2]), we'd calculate:

• ( f(0) = -2(0)^2 + 4(0) + 1 = 1 )
• ( f(2) = -2(2)^2 + 4(2) + 1 = 1 )

So, while we found a local maximum at ( x = 1 ), we should compare it to the values at the edges. Both endpoints give us ( 1 ), showing that the biggest value for ( f(x) ) on the interval ([0, 2]) is ( f(1) = 3 ), and the minimum values are at both ends with a value of ( 1 ).

Conclusion:

The First Derivative Test is a powerful way to find local maxima and minima in calculus. By following steps like finding the derivative, identifying critical points, testing intervals, and checking the sign changes of the derivative, we can get important information. This isn’t just about math; it helps solve real problems in business, physics, and engineering.

Practicing these steps with different functions will help you understand how derivatives show whether functions are increasing or decreasing. It’ll give you the tools you need to tackle not only school work but also real-world problems in many fields.