How do you find the exact relative maximum and minimum of the polynomial function of #f(x) =2x^33x^212x#?
Maximum
Minimum
Since the graph is not limited by a definition, we can just look for the extremal points through the dervivate of the function.
From here we can take several paths. Such as either use the quadratic equation or factorize to find the zero points. This will give us.
So by that, we can say that.
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To find the exact relative maximum and minimum of the polynomial function ( f(x) = 2x^3  3x^2  12x ), we first need to find the critical points by taking the derivative of the function and setting it equal to zero.

Find the derivative of ( f(x) ): ( f'(x) = 6x^2  6x  12 ).

Set the derivative equal to zero and solve for ( x ) to find the critical points: ( 6x^2  6x  12 = 0 ). Simplifying, we get ( x^2  x  2 = 0 ). Factoring, we get ( (x + 1)(x  2) = 0 ). So, the critical points are ( x = 1 ) and ( x = 2 ).

To determine if these critical points are relative maxima or minima, we use the second derivative test.
a. Find the second derivative of ( f(x) ): ( f''(x) = 12x  6 ).
b. Evaluate ( f''(1) ) and ( f''(2) ) to determine the concavity at the critical points. ( f''(1) = 12(1)  6 = 18 ) (negative, so concave down at ( x = 1 ), potential relative maximum). ( f''(2) = 12(2)  6 = 18 ) (positive, so concave up at ( x = 2 ), potential relative minimum).

Therefore, the function has a relative maximum at ( x = 1 ) and a relative minimum at ( x = 2 ).
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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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