What are the absolute extrema of #f(x)=2x^2 - 8x + 6 in[0,4]#?

Answer 1

#6# and #-2#

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.

We begin by evaluating the endpoints of the interval; in our case, that means finding #f(0)# and #f(4)#: #f(0)=2(0)^2-8(0)+6=6# #f(4)=2(4)^2-8(4)+6=6# Note that #f(0)=f(4)=6#.
Next, find the derivative: #f'(x)=4x-8->#using the power rule And find the critical points; i.e. the values for which #f'(x)=0#: #0=4x-8# #x=2# Evaluate the critical points (we only have one, #x=2#): #f(2)=2(2)^2-8(2)+6=-2#
Finally, determine the extrema. We see that we have a maximum at #f(x)=6# and a minimum at #f(x)=-2#; and since the question is asking what the absolute extrema are, we report #6# and #-2#. If the question was asking where the extrema occur, we would report #x=0#, #x=2#, and #x=4#.
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Answer 2

To find the absolute extrema of ( f(x) = 2x^2 - 8x + 6 ) on the interval [0, 4], follow these steps:

  1. Find the critical points of ( f(x) ) within the interval.
  2. Evaluate ( f(x) ) at the critical points and at the endpoints of the interval.
  3. 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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Answer from HIX Tutor

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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