Question

Is `f(x)=(4x^3-2x^2-x-3)/(x-2)` increasing or decreasing at `x=0`?

Answer 1

increasing at x = 0

Explanation:

To determine if a function is increasing / decreasing at x = a , we evaluate f'(a).

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

• If f'(a) < 0 , then f(x) is decreasing at x = a
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differentiate f(x) 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`
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g(x)`=4x^3-2x^2-x-3 rArr g'(x) = 12x^2 -4x - 1`

and h(x) = x-2 `rArr h'(x) = 1`
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substitute these values into f'(x)

f'(x) `= ((x-2)(12x^2-4x-1) - (4x^3-2x^2-x-3).1)/(x-2)^2`

and f'(0)`= ((-2)(-1) - (-3))/(-2)^2 = 5/4`

Since f'(0) > 0 , then f(x) is increasing at x = 0
graph{(4x^3-2x^2-x-3)/(x-2) [-10, 10, -5, 5]}