How do you find the relative extrema of #f(x)=3x^55x^3#?
Please see the explanation below
The function is
Calculate the first derivative
That is,
The solutions to this equation are
Let's build a variation chart
graph{3x^55x^3 [10, 10, 5, 5]}
By signing up, you agree to our Terms of Service and Privacy Policy
To find the relative extrema of ( f(x) = 3x^5  5x^3 ), you first need to find its critical points by taking the derivative of the function and setting it equal to zero. Then, you can determine whether these critical points correspond to relative maxima or minima by analyzing the sign of the second derivative at those points.

Take the derivative of ( f(x) ) with respect to ( x ) to find ( f'(x) ). [ f'(x) = 15x^4  15x^2 ]

Set ( f'(x) ) equal to zero and solve for ( x ) to find the critical points. [ 15x^4  15x^2 = 0 ] [ 15x^2(x^2  1) = 0 ] [ x^2(x+1)(x1) = 0 ]
So, the critical points are ( x = 1, 0, ) and ( x = 1 ).

Determine the sign of the second derivative ( f''(x) ) at each critical point to classify the relative extrema. [ f''(x) = 60x^3  30x ]

Plug the critical points into ( f''(x) ) and analyze the sign:
 ( x = 1 ): ( f''(1) = 90 ) (negative) implies a relative maximum.
 ( x = 0 ): ( f''(0) = 0 ) (indeterminate).
 ( x = 1 ): ( f''(1) = 30 ) (positive) implies a relative minimum.
Therefore, the relative maximum occurs at ( x = 1 ), and the relative 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.
 What is the maximum value that the graph of #sin^4x+cos^4x#?
 How do you know a function is increasing?
 Using the principle of the meanvalue theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem #f(x) = 1/(x1)#; [3, 0]?
 How do you find the extrema for #f(x) = x^2 +2x  4# for [1,1]?
 Is #f(x)=(x^26x12)/(x+2)# increasing or decreasing at #x=1#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7