How do you know a function is decreasing or increasing at #x=1# given the function #4x^2-9x#?

Answer 1
Usually what people mean when they ask this question is "Is the slope of the tangent line at #x=1# positive of negative"? Or "Is the rate of change at #x=1# positive or negative"?
For #f(x)=4x^2-9x#, the derivative is #f'(x)=8x-9#
Whether we think of the derivative as the slope of the tangent line or the rate of change, it is clear that when #x=1#, the derivative is negative. (#f'(1)=8(1)-9=-1#)
This is generally explained by saying that, at #x=1#, the function is decreasing at a rate of 1 (#f# unit) / (#x# unit).
(There is a bit of a conflict in terminology here. A function is constant at a single value of #x#. It is neither increasing nor decreasing. But in an interval containing a single value of #x#, the terms "increasing" and "decreasing" do apply.)
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Answer 2

To determine whether a function is increasing or decreasing at a particular point ( x = 1 ) given the function ( f(x) = 4x^2 - 9x ), you need to examine the sign of the derivative of the function at that point.

  1. Find the derivative of the function ( f(x) ) with respect to ( x ) using the power rule: ( f'(x) = 8x - 9 ).

  2. Evaluate the derivative at ( x = 1 ): ( f'(1) = 8(1) - 9 = 8 - 9 = -1 ).

  3. If ( f'(1) > 0 ), the function is increasing at ( x = 1 ). If ( f'(1) < 0 ), the function is decreasing at ( x = 1 ).

Since ( f'(1) = -1 < 0 ), the function ( f(x) = 4x^2 - 9x ) is decreasing at ( x = 1 ).

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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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