# Derivatives and concavity

Find the derivative of f(x)= (3-4x^2)/(x^2-x-6) and then determine intervals of increase/decrease,local max and min values and points of concavity and inflection. Also, please tell me that if the derivative of the function is (4x^2+42x+3)/(x^2-x-6) then do I use the denominator or numerator to determine whether the function is increasing or decreasing?

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#### Solution Preview

f(x)= (3-4x^2)/(x^2-x-6)

f' = (-8x) /(x^2-x-6) - (3-4x^2)(2x-1) / (x^2-x-6)^2

= [-8x*(x^2-x-6) - (3-4x^2)(2x-1)] / (x^2-x-6)^2

= [(-8x^3 + 8x^2+48x) + (8x^3-4x^2-6x+3)] / (x^2-x-6)^2

= (4x^2+42x+3)/(x^2-x-6)^2

we will study when does f' = 0

Since (x^2-x-6)^2 >= 0, we only need to care about the numerator.

Set 4x^2+42x+3 =0

Then x = [-42 + ...

#### Solution Summary

This shows how to find the derivative and then determine intervals of increase/decrease,local max and min values and points of concavity and inflection.