How do you find the critical points for #f(x) = (x^210x)^4# and the local max and min?
three critical points:
Relative minimum at
Relative maximum at
To find relative maximum and minimum, we can do a sign test.
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To find the critical points for ( f(x) = (x^2  10x)^4 ) and the local maximum and minimum, follow these steps:

Compute the first derivative of the function ( f(x) ).

Find the values of ( x ) where the derivative equals zero or is undefined.

Test the critical points by evaluating the second derivative at each point.

Determine the nature of each critical point:
a. If the second derivative is positive, it's a local minimum. b. If the second derivative is negative, it's a local maximum. c. If the second derivative is zero or undefined, the test is inconclusive.

Compute ( f(x) ) at each critical point to find the corresponding local maximum or minimum values.
Let's proceed with the steps:

Compute the first derivative of ( f(x) ): [ f'(x) = 4(x^2  10x)^3 \cdot (2x  10) ]

Find the critical points by setting the first derivative equal to zero: [ 4(x^2  10x)^3 \cdot (2x  10) = 0 ]
This gives us critical points at ( x = 0 ) and ( x = 5 ).

Test the critical points by evaluating the second derivative: [ f''(x) = 4 \cdot 3(x^2  10x)^2 \cdot (2x  10) + 4(x^2  10x)^3 \cdot 2 ]

Evaluate ( f''(x) ) at ( x = 0 ) and ( x = 5 ):
 At ( x = 0 ):
 ( f''(0) = 4 \cdot 3(0  0)^2 \cdot (2 \cdot 0  10) + 4(0  0)^3 \cdot 2 = 0 ) (inconclusive)
 At ( x = 5 ):
 ( f''(5) = 4 \cdot 3(5^2  10 \cdot 5)^2 \cdot (2 \cdot 5  10) + 4(5^2  10 \cdot 5)^3 \cdot 2 > 0 ) (positive)
 At ( x = 0 ):

Determine the nature of each critical point:
 At ( x = 0 ), the nature is inconclusive.
 At ( x = 5 ), it's a local minimum.

Compute ( f(x) ) at ( x = 0 ) and ( x = 5 ):
 At ( x = 0 ): ( f(0) = (0^2  10 \cdot 0)^4 = 0 )
 At ( x = 5 ): ( f(5) = (5^2  10 \cdot 5)^4 = 25^4 )
Therefore, the critical points for ( f(x) ) are ( x = 0 ) and ( x = 5 ), and ( f(x) ) has a local minimum at ( x = 5 ).
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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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