Question

Is `f(x)=(2x^3+2x^2+2x+1)/(x-3)` increasing or decreasing at `x=2`?

Answer 1

The function is decreasing.

Explanation:

You need to compute the first derivative in the given point. If it is positive, the function is increasing, if it is negative, the function is decreasing.

So, we have that

`f'(x)=\frac{(6x^2+4x+2)(x-3) - (2x^3+2x^2+2x+1)}{(x-3)^2}`
`=\frac{4x^3-16x^2-12x-7}{(x-3)^2}`

Evaluate `f'(2)`:

#=\frac{4*2^3-16*2^2-12*2-7}{(2-3)^2} =4*8-16*4-24-7#
`=32-64-24-7=-63`

So, the function is decreasing.