How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3 + 4x^2  5x#?
minimum:
maximum:
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To find the exact relative maximum and minimum of the polynomial function ( f(x) = x^3 + 4x^2  5x ), we first need to find the critical points by taking the derivative of the function and setting it equal to zero. Then, we determine whether these critical points correspond to relative maximum or minimum by analyzing the behavior of the function around those points using the second derivative test.

Find the derivative of ( f(x) ) with respect to ( x ):
( f'(x) = 3x^2 + 8x  5 )

Set the derivative equal to zero to find critical points:
( 3x^2 + 8x  5 = 0 )

Solve this quadratic equation for ( x ) to find the critical points.

Once you find the critical points, evaluate the second derivative of ( f(x) ), ( f''(x) ).

Plug the critical points into ( f''(x) ). If ( f''(x) > 0 ) at a critical point, then it's a relative minimum. If ( f''(x) < 0 ), it's a relative maximum.

Determine the ( y )values of the relative maximum and minimum by plugging the corresponding ( x )values back into the original function ( f(x) = x^3 + 4x^2  5x ).
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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.
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