Discover The Significance Of Points Of Inflection In Mathematics And Science

How to find 1st Derivative Test

find f'(x)f'(x) = 0 sign chart – f'(x)

The first derivative test is a technique used to determine the relative extremum (maximum or minimum) of a function. To use this test, you need to:

1. Take the first derivative of the function
2. Set the derivative equal to zero and solve for x. These values of x are called critical points.
3. Determine the sign of the derivative to the left and right of each critical point.
4. Use a chart or graph to determine whether the sign changes from positive to negative or negative to positive.
5. If the sign changes from positive to negative at a critical point, then that point is a local maximum. If the sign changes from negative to positive, then that point is a local minimum.

Here’s an example:

Let’s take the function f(x) = x^3 – 6x^2 + 9x + 2.

1. The first derivative of this function is f'(x) = 3x^2 – 12x + 9.
2. Set f'(x) equal to zero: 3x^2 – 12x + 9 = 0. Solve for x: x = 1 and x = 3.
3. Determine the sign of f'(x) to the left and right of each critical point:

– To the left of x=1, f'(x) is negative. To the right of x=1, f'(x) is positive.
– To the left of x=3, f'(x) is positive. To the right of x=3, f'(x) is negative.

4. Use a chart or graph to determine whether the sign changes from positive to negative or negative to positive. From the above step, we found out that the sign changes from negative to positive at x=1 and from positive to negative at x=3.
5. Therefore, x=1 is a local minimum and x=3 is a local maximum of the function.

So, using these steps, we can find the relative extrema of a function using the first derivative test.

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