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

Is #f(x)=(x+3)(x-2)(x-2)# increasing or decreasing at #x=-2#?

Answer 1

Aurora Alvarado

increasing at x = - 2

First, use FOIL to expand the function's brackets. Next, differentiate f(x) and verify its value.

(x + 3 )(#x^2 - 4x + 4 ) = x^3 - 2x^2 - 8x #

f(x) = #x^3 - x^2 - 8x + 12#

#rArr f'(x) = 3x^2 - 2x - 8#

# f'(-2) = 3(-2)^2 -2(-2) -8 = 12 + 4 - 8 = 8 > 0#

At x = -2, f(x) is increasing since f'(-2) > 0.

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Answer 2

Luke Alvarez

To determine whether ( f(x) = (x+3)(x-2)(x-2) ) is increasing or decreasing at ( x = -2 ), we can analyze the sign of the derivative of the function at that point.

( f'(x) = (x-2)(x-2) + (x+3)(2x-4) = (x-2)^2 + (x+3)(2x-4) )

Evaluating ( f'(-2) ):

( f'(-2) = (-2-2)^2 + (-2+3)(2(-2)-4) )

( f'(-2) = ( -4)^2 + (1)(-8) )

( f'(-2) = 16 - 8 )

( f'(-2) = 8 )

Since ( f'(-2) > 0 ), the function is increasing 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.