Is #f(x)=(3x^3-2x^2-2x+5)/(x+2)# increasing or decreasing at #x=3#?

Answer 1

#f'(x) =6x - 8 + 23/(x+2)^2#
and
#f'(3) = 273/25=10+23/25=10.92#

increasing

given

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

proceed by splitting up

#3x^3 - 2x^2 -2x + 5# by #x+2#

to acquire

#f(x) = 3x^2 - 8x +14 -23/(x+2)#

To obtain the first derivative, find

#f '(x) = 6x - 8+ 23/(x+2)^2#

evaluate

#f '(3) = 6(3)-8+23/(3+2)^2 = 10.92#
which indicates INCREASING at #x=3#
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Answer 2

To determine whether ( f(x) = \frac{{3x^3 - 2x^2 - 2x + 5}}{{x + 2}} ) is increasing or decreasing at ( x = 3 ), we can use the first derivative test. First, find the derivative of ( f(x) ) with respect to ( x ) using the quotient rule. Then, evaluate the derivative at ( x = 3 ). If the derivative is positive, the function is increasing at that point. If it's negative, the function is decreasing.

( f'(x) = \frac{{(9x^2 - 4x - 2)(x + 2) - (3x^3 - 2x^2 - 2x + 5)}}{{(x + 2)^2}} )

After simplifying, ( f'(x) = \frac{{3x^3 - 2x^2 - 2x + 5 - 9x^2 - 18x - 4x - 8}}{{(x + 2)^2}} )

( f'(x) = \frac{{3x^3 - 11x^2 - 24x - 3}}{{(x + 2)^2}} )

Evaluate ( f'(3) ):

( f'(3) = \frac{{3(3)^3 - 11(3)^2 - 24(3) - 3}}{{(3 + 2)^2}} )

( f'(3) = \frac{{81 - 99 - 72 - 3}}{{25}} )

( f'(3) = \frac{{-93}}{{25}} )

Since ( f'(3) ) is negative, the function ( f(x) ) is 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.

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.

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.

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.

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