How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3 - 6x^2 + 9x +1#?

Question

Isaiah Allen · Accepted Answer

#(1, 5)# Maximum Point
#(3, 1)# Minimum Point

Given #f(x)=x^3-6x^2+9x+1#

Find the first derivative #f' (x)# then equate to zero then solve for the point.

#f(x)=x^3-6x^2+9x+1#

#f' (x)=3x^2-12x+9#

#f' (x)=3x^2-12x+9=0#

#x^2-4x+3=0#

#(x-1)(x-3)=0#

#x-1=0# and #x-3=0#

#x=1# and #x=3#

At #x=1#

#f(x)=x^3-6x^2+9x+1#

#f(1)=1^3-6(1)^2+9(1)+1#

#f(1)=1-6+9+1#

#f(1)=5#

At #x=3#

#f(x)=x^3-6x^2+9x+1#

#f(1)=(3)^3-6(3)^2+9(3)+1#

#f(1)=27-54+27+1#

#f(1)=1#

So the points are #(1, 5)# and #(3, 1)#

#(1, 5)# Maximum Point
#(3, 1)# Minimum Point

God bless....I hope the explanation is useful.

Adam Adkins · Answer

undefined To find the exact relative maximum and minimum of the polynomial function $ f(x) = x^3 - 6x^2 + 9x + 1 $, follow these steps:

1. Find the critical points by taking the derivative of $ f(x) $ and setting it equal to zero:  
   $ f'(x) = 3x^2 - 12x + 9 $  
   Setting $ f'(x) = 0 $ gives:  
   $ 3x^2 - 12x + 9 = 0 $  
   Solve this quadratic equation to find the critical points.

2. Once you have the critical points, evaluate $ f(x) $ at each critical point and at the endpoints of the interval you're interested in (if any).

3. The point where the function changes from increasing to decreasing (or vice versa) is a relative maximum or minimum, respectively. Identify these points among the critical points and endpoints.

4. Evaluate $ f(x) $ at these points to confirm which are relative maximum or minimum.

5. The highest point among the relative maximums is the absolute maximum, and the lowest point among the relative minimums is the absolute minimum.