Answer to Question #212390 in Calculus for Kenneth

Question #212390

(a) find and classify the critical points of the functions f(x) = 2x^3 + 3x^2 - 12 x +1 into maximum, minimum and inflection points as appreciate.

(b) The sum of two positive numbers is S. find the maximum value of their product.


1
Expert's answer
2021-07-01T13:38:06-0400

(a)


"f(x) = 2x^3 + 3x^2 - 12 x +1"

Domain: "(-\\infin, \\infin)" as polynomial.

Find the first derivative with respect to "x"


"f'(x)=( 2x^3 + 3x^2 - 12 x +1)'=6x^2+6x-12"

Find the critical number(s)


"f'(x)=0=>6x^2+6x-12=0"

"6(x-1)(x+2)=0"

Critical numbers: "-2, 1."

By the First Derivative Test

If "x<-2," then "f'(x)>0, f(x)" increases.

If "-2<x<1," then "f'(x)<0, f(x)" decreases.

If "x>1," then "f'(x)>0, f(x)" increases.



"f(-2)=2(-2)^3 + 3(-2)^2 - 12 (-2) +1=21"

"f(1)=2(1)^3 + 3(1)^2 - 12 (1) +1=-6"

The function "f" has a local maximum with value of "21" at "x=-2."

The function "f" has a local minimum with value of "-6" at "x=1."


(b) Let "x=" the first number, "x>0." Then the second number will be "S-x."

Given "S-x>0."

Their product will be


"f(x)=x(S-x), 0<x<S"

Find the first derivative with respect to "x"


"f'(x)=( x(S-x))'=S-2x"

Find the critical number(s)


"f'(x)=0=>S-2x=0"

"x=\\dfrac{1}{2}S"

Critical number: "\\dfrac{1}{2}S ."

By the First Derivative Test

If "x<\\dfrac{1}{2}S," then "f'(x)>0, f(x)" increases.


If "x>\\dfrac{1}{2}S," then "f'(x)<0, f(x)" decreases.


"f(\\dfrac{1}{2}S)=\\dfrac{1}{2}S(S-\\dfrac{1}{2}S)=\\dfrac{1}{4}S^2"

The function "f" has a local maximum with value of "\\dfrac{1}{4}S^2" at "x=\\dfrac{1}{2}S."

Since the function "f" has the only extremum, then the function "f" has the absolute maximum with value of "\\dfrac{1}{4}S^2" at "x=\\dfrac{1}{2}S" for "0<x<S."


The maximum value of the product "\\dfrac{1}{4}S^2" will be if we take two equal positive numbers


"I\\ number=\\dfrac{1}{2}S=II\\ number"

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