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

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

Answer 1

Paisley Johnson

#f(x)=(-x-4)^2+3x^2-3x# is decreasing at #x=-1#

From the given equation, obtain the first derivative and then substitute the value of #x=-1# #f(x)=(-x-4)^2+3x^2-3x#

#f(x)=x^2+8x+16+3x^2-3x#

#f(x)=4x^2+5x+16#

#f'(x)=8x+5#

#f'(-1)=8(-1)+5#

#f'(-1)=-3" "#this means decreasing

May God bless you all. I hope this explanation helps.

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Answer 2

Emilia Aguilar

To determine if the function ( f(x) = (-x - 4)^2 + 3x^2 - 3x ) is increasing or decreasing at ( x = -1 ), we need to evaluate the sign of the derivative of the function at that point.

First, find the derivative of ( f(x) ) with respect to ( x ):

( f'(x) = 2(-x - 4)(-1) + 6x - 3 )

Now, evaluate ( f'(-1) ):

( f'(-1) = 2(-(-1) - 4)(-1) + 6(-1) - 3 )

( f'(-1) = 2(-(-1) - 4)(-1) - 6 - 3 )

( f'(-1) = 2(-1 + 4)(-1) - 6 - 3 )

( f'(-1) = 2(3)(-1) - 6 - 3 )

( f'(-1) = -6 - 6 - 3 )

( f'(-1) = -15 )

Since the derivative ( f'(-1) = -15 ) is negative, the function ( f(x) ) is decreasing at ( x = -1 ).

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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.