# How do you find the exact relative maximum and minimum of the polynomial function of #f(x)=4x-x^2#?

At

#dy/dx=0# and#(d^2y)/(dx^2)<0#

Hence the function has a maximum at

graph{4x-x^2 [-10, 10, -5, 5]}

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To find the exact relative maximum and minimum of the polynomial function ( f(x) = 4x - x^2 ):

- Find the critical points by setting the derivative equal to zero and solving for ( x ).
- Determine the nature of each critical point by evaluating the second derivative at each point.
- Confirm whether the critical points are relative maximum, minimum, or points of inflection.

Let's start with finding the derivative of ( f(x) ): ( f'(x) = 4 - 2x ).

Setting ( f'(x) ) equal to zero: ( 4 - 2x = 0 ), ( 2x = 4 ), ( x = 2 ).

Now, find the second derivative of ( f(x) ): ( f''(x) = -2 ).

Since ( f''(x) ) is constant and negative, the critical point at ( x = 2 ) is a relative maximum.

Therefore, the exact relative maximum of ( f(x) ) is at ( x = 2 ), and the maximum value is ( f(2) = 4 ).

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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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