How do you find the exact relative maximum and minimum of the polynomial function of #f(x)=x^33x+6#?
Relative maximum:
Test the critical numbers
Use either the first or second derivative test to see that
Determine the lowest and highest numbers.
By signing up, you agree to our Terms of Service and Privacy Policy
To find the exact relative maximum and minimum of the polynomial function ( f(x) = x^3  3x + 6 ), you can follow these steps:
 Find the critical points by setting the derivative of ( f(x) ) equal to zero and solving for ( x ).
 Determine the nature of these critical points (whether they correspond to relative maximum, minimum, or inflection points) using the second derivative test.
 Evaluate ( f(x) ) at the critical points to find the corresponding function values, which will give you the relative maximum and minimum.
Let's proceed with the calculations:
 Find the derivative of ( f(x) ): [ f'(x) = 3x^2  3 ]
Setting ( f'(x) = 0 ): [ 3x^2  3 = 0 ] [ x^2  1 = 0 ] [ (x  1)(x + 1) = 0 ]
So, the critical points are ( x = 1 ) and ( x = 1 ).

Find the second derivative of ( f(x) ): [ f''(x) = 6x ]

Evaluate ( f''(x) ) at the critical points: [ f''(1) = 6(1) = 6 > 0 ] [ f''(1) = 6(1) = 6 < 0 ]
By the second derivative test:
 At ( x = 1 ), since ( f''(1) > 0 ), ( f(x) ) has a relative minimum at ( x = 1 ).
 At ( x = 1 ), since ( f''(1) < 0 ), ( f(x) ) has a relative maximum at ( x = 1 ).
Now, evaluate ( f(x) ) at these critical points: [ f(1) = (1)^3  3(1) + 6 = 1  3 + 6 = 4 ] [ f(1) = (1)^3  3(1) + 6 = 1 + 3 + 6 = 8 ]
So, the relative minimum is at ( (1, 4) ) and the relative maximum is at ( (1, 8) ).
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 are the points of inflection of #f(x)=x^5  2x^3 +4x#?
 For what values of x is #f(x)=x^2+xe^x# concave or convex?
 How do you use the second derivative test to find the relative maxima and minima of the given #f(x)= x^4  (2x^2) + 3#?
 What are the points of inflection of #f(x)=1/(5x^2+3) #?
 How do you find the first and second derivative of #(ln(x))^2#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7