What are the absolute extrema of #f(x)=2x^2 - 8x + 6 in[0,4]#?
Absolute extrema (the min. and max. values of a function over an interval) can be found by evaluating the endpoints of the interval and the points where the derivative of the function equals 0.
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To find the absolute extrema of ( f(x) = 2x^2 - 8x + 6 ) on the interval [0, 4], follow these steps:
- Find the critical points of ( f(x) ) within the interval.
- Evaluate ( f(x) ) at the critical points and at the endpoints of the interval.
- Identify the maximum and minimum values among these.
Step 1: Find the critical points: To find the critical points, take the derivative of ( f(x) ) and set it equal to zero. ( f'(x) = 4x - 8 ) Set ( f'(x) = 0 ) and solve for ( x ): ( 4x - 8 = 0 ) ( x = 2 ) So, ( x = 2 ) is the only critical point within the interval [0, 4].
Step 2: Evaluate ( f(x) ) at the critical point and endpoints: ( f(0) = 2(0)^2 - 8(0) + 6 = 6 ) ( f(2) = 2(2)^2 - 8(2) + 6 = -6 ) ( f(4) = 2(4)^2 - 8(4) + 6 = 6 )
Step 3: Identify the maximum and minimum values: The maximum value occurs at ( x = 0 ) and ( x = 4 ), where ( f(x) = 6 ). The minimum value occurs at ( x = 2 ), where ( f(x) = -6 ).
Therefore, the absolute maximum value is 6, occurring at ( x = 0 ) and ( x = 4 ), and the absolute minimum value is -6, occurring at ( x = 2 ).
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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