How do you find the exact relative maximum and minimum of the polynomial function of # f(x) = –2x^3 + 6x^2 + 18x –18 #?
The relative maximum is
The polynomial function is
The first derivative is
Therefore,
We can make a variation chart
The second derivative is
graph{2x^3+6x^2+18x18 [74.1, 74.1, 37.03, 37]}
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To find the exact relative maximum and minimum of the polynomial function ( f(x) = 2x^3 + 6x^2 + 18x  18 ), you first need to find its critical points by taking the derivative and setting it equal to zero. Then, you can use the second derivative test to determine whether each critical point corresponds to a relative maximum, minimum, or neither.

Find the derivative of ( f(x) ): [ f'(x) = 6x^2 + 12x + 18 ]

Set ( f'(x) ) equal to zero and solve for ( x ): [ 6x^2 + 12x + 18 = 0 ] [ 2x^2 + 4x + 6 = 0 ] [ x^2 + 2x + 3 = 0 ] [ (x  3)(x  1) = 0 ]
The critical points are ( x = 3 ) and ( x = 1 ).

Find the second derivative ( f''(x) ): [ f''(x) = 12x + 12 ]

Evaluate ( f''(x) ) at the critical points: [ f''(3) = 12(3) + 12 = 36 + 12 = 24 ] (negative, so it's a local maximum) [ f''(1) = 12(1) + 12 = 12 + 12 = 24 ] (positive, so it's a local minimum)

Therefore, the function has a relative maximum at ( x = 3 ) and a relative minimum at ( x = 1 ).
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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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