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

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

Answer 1

Ezekiel Alexander

#f'(-1)=-9<0# so the function is decreasing at #x=-1#

Given #f(x)=(x+3)(x-2)(x+4)# expanding the Right-Hand side:

#(x+3)(x-2)(x+4)=(x^2+3x-2x-6)(x+4)=(x^2+x-6)(x+4)=x^3+x^2-6x+4x^2+4x-24#

#f'(x)=x^3+5x^2-2x-24#

#f'(x)=3x^2+10x-2# and

#f'(-1)=3-10-2=-9<0#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Scarlett Allison

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

Find the derivative of the function ( f(x) ) using the product rule. [ f'(x) = (x-2)(x+4) + (x+3)(x+4) + (x+3)(x-2) ]
Evaluate the derivative at ( x = -1 ). [ f'(-1) = (-1-2)(-1+4) + (-1+3)(-1+4) + (-1+3)(-1-2) ] [ f'(-1) = (-3)(3) + (2)(3) + (2)(-3) ] [ f'(-1) = -9 + 6 - 6 ] [ f'(-1) = -9 ]
Since the derivative ( f'(-1) = -9 ) is negative, the function ( f(x) ) is decreasing at ( x = -1 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.