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

The function is decreasing.

Calculate the first derivative at the specified point; a positive value indicates an increasing function, and a negative value indicates a decreasing function.

Thus, we possess that

Thus, the function is getting smaller.

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

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

[ f'(x) = \frac{{(x - 3)(6x^2 + 4x + 2) - (2x^3 + 2x^2 + 2x + 1)(1)}}{{(x - 3)^2}} ]

Next, plug in ( x = 2 ) into ( f'(x) ) to find the sign:

[ f'(2) = \frac{{(2 - 3)(6(2)^2 + 4(2) + 2) - (2(2)^3 + 2(2)^2 + 2(2) + 1)(1)}}{{(2 - 3)^2}} ]

[ f'(2) = \frac{{(-1)(24 + 8 + 2) - (16 + 8 + 4 + 1)}}{{(-1)^2}} ]

[ f'(2) = \frac{{-34 - 29}}{{1}} = -63 ]

Since ( f'(2) ) is negative, ( f(x) ) is decreasing at ( x = 2 ).

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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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- How do you find the critical numbers of #y = (x^2-4)/(x^2-2x)#?
- How do I find the numbers #c# that satisfy the Mean Value Theorem for #f(x)=3x^2+2x+5# on the interval #[-1,1]# ?
- Is #f(x)=-2x^3+4x^2-3x-1# increasing or decreasing at #x=0#?
- Is #f(x)=cosx+tanx-sinx# increasing or decreasing at #x=pi/6#?

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