What are the global and local extrema of #f(x)=x^2 2x +3# ?
Determine the critical points of the function, by solving the equation:
Evaluate the second derivative in this point:
As the second derivative is positive this critical point is a local minimum, and the value of the function at the minimum is:
Now consider the function:
this is a perfect square:
and then:
On the other hand:
so the function is not bounded and can have no absolute maximum.
graph{x^22x+3 [8.375, 11.625, 0.48, 10.48]}
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To find the global and local extrema of ( f(x) = x^2  2x + 3 ), we first need to find the critical points by setting the derivative equal to zero and then determine whether these points correspond to maxima, minima, or neither.

Find the derivative of ( f(x) ): [ f'(x) = 2x  2 ]

Set the derivative equal to zero and solve for ( x ) to find critical points: [ 2x  2 = 0 ] [ x = 1 ]

To determine whether ( x = 1 ) corresponds to a maximum, minimum, or neither, we can use the second derivative test. [ f''(x) = 2 ]
Since ( f''(1) = 2 > 0 ), the function has a local minimum at ( x = 1 ).
 Since there are no other critical points, ( x = 1 ) is the only critical point of ( f(x) ), thus the local minimum is also the global minimum.
Therefore, the global and local minimum of ( f(x) = x^2  2x + 3 ) is ( (1, 2) ). There are no global or local maxima.
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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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