How do use the first derivative test to determine the local extrema #f(x) = x / (x^2+1)#?
See the explanation.
By signing up, you agree to our Terms of Service and Privacy Policy
To use the first derivative test to determine the local extrema of ( f(x) = \frac{x}{x^2 + 1} ), follow these steps:
 Find the first derivative of the function, ( f'(x) ).
 Set ( f'(x) ) equal to zero and solve for ( x ). These points are potential critical points where the function may have local extrema.
 Determine the sign of ( f'(x) ) in the intervals defined by the critical points found in step 2.
 If the sign of ( f'(x) ) changes from positive to negative at a critical point, then that point is a local maximum. If the sign changes from negative to positive, then it is a local minimum.
 If ( f'(x) ) does not change sign at a critical point, then the test is inconclusive at that point.
Now, applying these steps to the function ( f(x) = \frac{x}{x^2 + 1} ):

Find ( f'(x) ) by using the quotient rule: [ f'(x) = \frac{(1)(x^2 + 1)  (2x)(x)}{(x^2 + 1)^2} ] [ f'(x) = \frac{x^2 + 1  2x^2}{(x^2 + 1)^2} ] [ f'(x) = \frac{1  x^2}{(x^2 + 1)^2} ]

Set ( f'(x) = 0 ) and solve for ( x ): [ 1  x^2 = 0 ] [ x^2 = 1 ] [ x = \pm 1 ]

We have two critical points, ( x = 1 ) and ( x = 1 ). Now, determine the sign of ( f'(x) ) in the intervals ( (\infty, 1) ), ( (1, 1) ), and ( (1, \infty) ):
 For ( x < 1 ), ( f'(x) ) is positive.
 For ( 1 < x < 1 ), ( f'(x) ) is negative.
 For ( x > 1 ), ( f'(x) ) is positive.

Since ( f'(x) ) changes from positive to negative at ( x = 1 ) and from negative to positive at ( x = 1 ), we have a local maximum at ( x = 1 ) and a local minimum at ( x = 1 ).
Therefore, the local maximum occurs at ( x = 1 ) and the local minimum occurs at ( x = 1 ).
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 find the critical point and determine whether it is a local maximum, local minimum, or neither for #f(x, y) = x^2 + 4x + y^2#?
 How do you determine all values of c that satisfy the mean value theorem on the interval [1, 4] for #f(x)=1/sqrt(x)#?
 How do you find the local max and min for # y = 3x^4 + 4x^3 – 12x^2 + 1#?
 What are the local extrema, if any, of #f (x) =2ln(x^2+3)x#?
 How do you find the critical points if #f'(x)=2x/(x+2)^3#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7