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

Apr 16, 2016

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$\textcolor{b l u e}{\text{ quotient rule }}$

If f(x)$= \frac{g \left(x\right)}{h \left(x\right)} \text{ then } f ' \left(x\right) = \frac{h \left(x\right) . g ' \left(x\right) - g \left(x\right) . h ' \left(x\right)}{h \left(x\right)} ^ 2$
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g(x)$= 4 {x}^{3} - 2 {x}^{2} - x - 3 \Rightarrow g ' \left(x\right) = 12 {x}^{2} - 4 x - 1$

and h(x) = x-2 $\Rightarrow h ' \left(x\right) = 1$
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substitute these values into f'(x)

f'(x) $= \frac{\left(x - 2\right) \left(12 {x}^{2} - 4 x - 1\right) - \left(4 {x}^{3} - 2 {x}^{2} - x - 3\right) .1}{x - 2} ^ 2$

and f'(0)$= \frac{\left(- 2\right) \left(- 1\right) - \left(- 3\right)}{- 2} ^ 2 = \frac{5}{4}$

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]}