What are the absolute extrema of # f(x)= 2 + x^2 in [2, 3]#?
graph{2+x^2 [9.19, 8.59, 0.97, 7.926]}
By signing up, you agree to our Terms of Service and Privacy Policy
To find the absolute extrema of ( f(x) = 2 + x^2 ) in the interval ([2, 3]), we first need to evaluate the function at the endpoints of the interval and at any critical points within the interval.

Evaluate ( f(x) ) at the endpoints: ( f(2) = 2 + (2)^2 = 2 + 4 = 6 ) ( f(3) = 2 + (3)^2 = 2 + 9 = 11 )

Find critical points by taking the derivative of ( f(x) ) and setting it equal to zero: ( f'(x) = 2x ) Setting ( f'(x) = 0 ) gives ( x = 0 ).

Evaluate ( f(x) ) at the critical point: ( f(0) = 2 + (0)^2 = 2 )
Comparing the values, we find that the absolute minimum is ( f(2) = 6 ) and the absolute maximum is ( f(3) = 11 ).
By signing up, you agree to our Terms of Service and Privacy Policy
To find the absolute extrema of ( f(x) = 2 + x^2 ) in the interval ([2, 3]), we need to evaluate the function at the endpoints of the interval as well as at any critical points within the interval.
The critical points occur where the derivative of the function is equal to zero or undefined.
The derivative of ( f(x) ) is ( f'(x) = 2x ).
Setting ( f'(x) ) equal to zero, we find the critical point to be ( x = 0 ).
Next, we evaluate ( f(x) ) at the endpoints and the critical point:
 At ( x = 2 ), ( f(2) = 2 + (2)^2 = 6 )
 At ( x = 0 ), ( f(0) = 2 + 0^2 = 2 )
 At ( x = 3 ), ( f(3) = 2 + 3^2 = 11 )
Comparing the values, the absolute maximum occurs at ( x = 3 ) where ( f(x) = 11 ), and the absolute minimum occurs at ( x = 0 ) where ( f(x) = 2 ).
By signing up, you agree to our Terms of Service and Privacy Policy
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.
 How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = 10x^4  2x^2 + 7x  1# has a root in the interval [3, 0]?
 Is #f(x)=(x^34x^22x4)/(x1)# increasing or decreasing at #x=2#?
 How do you find the maximum value of #f(x)=2sin(x)+cos(x)#?
 How do you find the relative extrema of #f(x)=3x^55x^3#?
 Is #f(x)=cos(x)# increasing or decreasing at #x=0#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7