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

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

Answer 1

Christopher Agresta

Increasing.

We compute the slope of the function by finding the first derivative. To do this we shall use the quotient rule:

#f'(x) = ((-4x-4)(2x^2+1) - 4x(-2x^2-4x-2))/((2x^2+1)^2)#

#f'(x) = (-8x^3-4x-8x^2-4 + 8x^3 + 16x^2 + 8x)/((2x^2+1)^2)#

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

#f'(-3) = (8(-3)^2 + 4(-3)-4)/(2(-3)^2+1)^2 = (72-16)/(19)^2 > 0#

#implies# function is increasing, as #f'(x) > 0#

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Answer 2

Isaiah Allen

To determine if ( f(x) = \frac{-2x^2 - 4x - 2}{2x^2 + 1} ) is increasing or decreasing at ( x = -3 ), we evaluate the derivative of the function at that point. If the derivative is positive, the function is increasing at that point. If the derivative is negative, the function is decreasing at that point.

The derivative of ( f(x) ) can be found using the quotient rule:

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

Evaluating ( f'(-3) ) will determine if the function is increasing or decreasing at ( x = -3 ).

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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.