# given: 1st derivative is

2(2-3x-2x^2)/(1+x^2)^2 (quotient rule)

I've found the stationary points to be x=1/2 and x=-2

Am I on the right track?

If so, I plan to plug in values to the L andR of them into the whole derivative so as to clasify them. Still on track?

Thanks

those are the correct values of x where the derivative is zero.

I don't know what you mean by the second part.

Thanks for the confirmation: The second part refers to deciding whether the zero points are minima or maxima.

## To determine whether the zero points are minima or maxima, you can use the first derivative test. The first derivative represents the rate of change of the function, so at the points where the derivative is zero, the function is neither increasing nor decreasing.

To apply the first derivative test, you need to evaluate the sign of the derivative on both sides of the zero points. If the sign of the derivative changes from positive to negative, then you have a local maximum at that point. If the sign changes from negative to positive, then you have a local minimum. If the sign does not change, then you have neither a minimum nor a maximum (a point of inflection).

To determine the sign of the derivative at a point, you can substitute a value greater than the point into the derivative and evaluate it. If the result is positive, then the derivative is positive at that point. Similarly, if the result is negative, then the derivative is negative at that point.

So, to classify the zero points as minima or maxima, plug in values to the left and right of each zero point into the derivative. Determine the sign of the derivative for each value and compare them.

For example, let's take x = 0 (to the left of x = 1/2). Plug 0 into the derivative:

2(2 - 3(0) - 2(0)^2) / (1 + (0)^2)^2 = 4 / 1 = 4

The result is positive, so the derivative is positive to the left of x = 1/2. Repeat this process for a value greater than x = 1/2 to the right of it. If the sign changes from positive to negative, then you have a local maximum. If not, you have neither a minimum nor a maximum.

Repeat this process for x = -2 as well.

I hope this clears up how to classify the zero points. Let me know if you have any more questions!