Question

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

Answer 1

increasing at x = 1

Explanation:

To check if function is increasing / decreasing , require to find the value of f'(1)

• If f'(1) > 0 then f(x) is increasing at x = 1

• If f'(1) < 0 then f(x) is decreasing at x = 1

Differentiate using the `color(blue)" Quotient rule "`

If f(x) `= g(x)/(h(x)) " then " f'(x) = (h(x)g'(x) - g(x)h'(x))/[h(x)]^2`

hence `f'(x) = ((x-3) d/dx(x^2-2x-2) - (x^2-2x-2) d/dx(x-3))/(x-3)^2`

`= ((x-3)(2x-2) - (x^2-2x-2).1)/(x-3)^2`

`f'(1) = ((1-3)(2-2) - (1-2-2))/(1-3)^2 = (0-(-3))/4 = 3/4`

`rArr " since " f'(1) > 0 " then f(x) is increasing at x = 1 "`
This can be seen on graph of function.
graph{(-x^2-2x-2)/(x-3) [-10, 10, -5, 5]}