How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3  6x^2 + 9x +1#?
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Find the first derivative
At
At
So the points are
God bless....I hope the explanation is useful.
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To find the exact relative maximum and minimum of the polynomial function ( f(x) = x^3  6x^2 + 9x + 1 ), follow these steps:

Find the critical points by taking the derivative of ( f(x) ) and setting it equal to zero:
( f'(x) = 3x^2  12x + 9 )
Setting ( f'(x) = 0 ) gives:
( 3x^2  12x + 9 = 0 )
Solve this quadratic equation to find the critical points. 
Once you have the critical points, evaluate ( f(x) ) at each critical point and at the endpoints of the interval you're interested in (if any).

The point where the function changes from increasing to decreasing (or vice versa) is a relative maximum or minimum, respectively. Identify these points among the critical points and endpoints.

Evaluate ( f(x) ) at these points to confirm which are relative maximum or minimum.

The highest point among the relative maximums is the absolute maximum, and the lowest point among the relative minimums is the absolute minimum.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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