Is #f(x)=(3x^3+x^2-2x-14)/(x-1)# increasing or decreasing at #x=2#?

Answer 1

At #x=2#, #f(x)=(3x^3+x^2-2x-14)/(x-1)# is increasing.

For any function, if #(df)/(dx)# is positive, it is increasing, if #(df)/(dx)# is negative, it is decreasing and if if #(df)/(dx)=0#, it is flat.
in the case of #f(x)=(3x^3+x^2-2x-14)/(x-1)#
#(df)/(dx)=((x-1)(9x^2+2x-2)-(3x^3+x^2-2x-14)xx1)/(x-1)^2#
= #(9x^3+2x^2-2x-9x^2-2x+2-3x^3-x^2+2x+14)/(x-1)^2#
= #(6x^3-8x^2-2x+16)/(x-1)^2#
And at #x=2#,
#(df)/(dx)=(6*2^3-8*2^2-2*2+16)/(2-1)^2#
= #(48-32-4+16)/1=28#
Hence, at #x=2#, #(3x^3+x^2-2x-14)/(x-1)# is increasing.
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Answer 2

To determine whether ( f(x) = \frac{{3x^3 + x^2 - 2x - 14}}{{x - 1}} ) is increasing or decreasing at ( x = 2 ), we can use the first derivative test. We'll find the derivative of ( f(x) ), then evaluate it at ( x = 2 ).

First, find the derivative ( f'(x) ) using the quotient rule:

[ f'(x) = \frac{{(x - 1) \cdot (9x^2 + 2x - 2) - (3x^3 + x^2 - 2x - 14) \cdot 1}}{{(x - 1)^2}} ]

Now, evaluate ( f'(x) ) at ( x = 2 ) to determine whether the function is increasing or decreasing at that point.

[ f'(2) = \frac{{(2 - 1) \cdot (9(2)^2 + 2(2) - 2) - (3(2)^3 + (2)^2 - 2(2) - 14) \cdot 1}}{{(2 - 1)^2}} ]

After calculating, if ( f'(2) > 0 ), then ( f(x) ) is increasing at ( x = 2 ). If ( f'(2) < 0 ), then ( f(x) ) is decreasing 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.

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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