Home > Calculus > Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

Is #f(x)=(x^3-4x^2-2x-4)/(x-1)# increasing or decreasing at #x=2#?

Answer 1

Increasing.

Express the ratio in the quotient remainder form #x^2-3x-5-9/(x-1)# #f' = 2x-3 + 9/(x-1)^2#. At x = 2, f' = 10 > 0. So, f is increasing at x = 2.

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Answer 2

Axel Alvarez

To determine if ( f(x) = \frac{x^3 - 4x^2 - 2x - 4}{x - 1} ) is increasing or decreasing at ( x = 2 ), we can evaluate the sign of the derivative at that point. If ( f'(x) > 0 ), the function is increasing, and if ( f'(x) < 0 ), the function is decreasing.

First, find the derivative ( f'(x) ) using the quotient rule:

[ f'(x) = \frac{(x - 1)(3x^2 - 8x - 2) - (x^3 - 4x^2 - 2x - 4)(1)}{(x - 1)^2} ]

Evaluate ( f'(2) ) to determine if the function is increasing or decreasing at ( x = 2 ). If ( f'(2) > 0 ), the function is increasing. If ( f'(2) < 0 ), the function is decreasing. If ( f'(2) = 0 ), the function has a horizontal tangent at that point.

After evaluating ( f'(2) ), we can determine whether the function is increasing or decreasing at ( x = 2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.