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

Answer 1

Increasing.

Calculate the first derivative when #x=-2#. If the value is negative, the function is decreasing at that point.If the value is positive, the value is increasing at that point.

First, divide the binomials to avoid using the tedious three-pronged product rule.

#f(x)=4x^3-16x^2+19x-6#
#f'(x)=12x^2-32x+19#
#f'(-2)=48+64+19#
#f'(-2)# is positive, so the function is increasing at this point.
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Answer 2

To determine whether ( f(x) = (x - 2)(2x - 3)(2x - 1) ) is increasing or decreasing at ( x = -2 ), we evaluate the sign of the derivative of ( f(x) ) at that point.

To find the derivative of ( f(x) ), we use the product rule:

[ f'(x) = (x - 2)'(2x - 3)(2x - 1) + (x - 2)(2x - 3)'(2x - 1) + (x - 2)(2x - 3)(2x - 1)' ]

[ f'(x) = (1)(2x - 3)(2x - 1) + (x - 2)(2)(2x - 1) + (x - 2)(2x - 3)(2) ]

[ f'(x) = 4x^2 - 10x + 3 ]

Now, evaluate ( f'(-2) ):

[ f'(-2) = 4(-2)^2 - 10(-2) + 3 ]

[ f'(-2) = 16 + 20 + 3 ]

[ f'(-2) = 39 ]

Since ( f'(-2) ) is positive, ( f(x) ) is increasing 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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