Question

For the function f(x)=2x^3-3x^2-22.5x+19

a. Find the critical points or explain, with a derivative, why there are no critical points.
b. Find the x and y coordinates of the absolute maximum and absolute minimum of f(x) in the interval [-4,5]. Be sure to show all the necessary supporting calculus
c. Find the x and y coordinates of the absolute max and absolute min of f(x) in the interval [0,3]. Be sure to show all the necessary supporting calculus.
d. Find all intervals where f(x) is increasing and all intervals where f9x0 is decreasing. Be sure to show supporting calculus.
e. Find all intervals where f(x) is concave up and all intervals where f(x) is concave down. Show supporting calculus

Answer 1

A.
`f(x)=2x^3-3x^2-22.5x+19\\f'(x)=6x^2-6x-22.5\\f'(x)=6x^2-6x-22.5=0\\4x^2-4x-15=0\\(2x+3)(2x-5)=0\\2x+3=0~|~2x-5=0\\2x=-3~|~2x=5\\x=-1.5~|~x=2.5\\x=-1.5,2.5\\\\f''(x)=12x-6\\12x-6=0\\12x=6\\x=0.5` - Relative maximum at x = -1.5. Relative minimum at x = 2.5. Point of inflection at x = 0.5.

B.
`f(x)=2x^3-3x^2-22.5x+19\\f'(x)=6x^2-6x-22.5\\f'(x)=6x^2-6x-22.5=0\\4x^2-4x-15=0\\(2x+3)(2x-5)=0\\2x+3=0~|~2x-5=0\\2x=-3~|~2x=5\\x=-1.5~|~x=2.5\\x=-1.5,2.5` - Absolute maximum at x = -1.5. Absolute minimum at x = 2.5. This applies for the interval [-4, 5].

C.
`f(x)=2x^3-3x^2-22.5x+19\\f(0)=2(0)^3-3(0)^2-22.5(0)+19\\f(0)=19\\\\f'(x)=6x^2-6x-22.5\\f'(x)=6x^2-6x-22.5=0\\4x^2-4x-15=0\\(2x+3)(2x-5)=0\\2x+3=0~|~2x-5=0\\2x=-3~|~2x=5\\x=-1.5~|~x=2.5\\x=2.5` - Absolute maximum at x = 0. Absolute minimum at x = 2.5. This applies for the interval [0, 3].

D.
`f(x)=2x^3-3x^2-22.5x+19\\f'(x)=6x^2-6x-22.5\\f'(x)=6x^2-6x-22.5=0\\4x^2-4x-15=0\\(2x+3)(2x-5)=0\\2x+3=0~|~2x-5=0\\2x=-3~|~2x=5\\x=-1.5~|~x=2.5\\x=-1.5,2.5` - Test:
`f(-2)=2x^3-3x^2-22.5x+19\\f(-2)=2(-2)^3-3(-2)^2-22.5(-2)+19\\f(-2)=2(-8)-3(4)-45+19\\f(-2)=-16-12-26\\f(-2)=-54<0\\\\f(0)=2(0)^3-3(0)^2-22.5(0)+19\\f(0)=19>0\\\\f(3)=2x^3-3x^2-22.5x+19\\f(3)=2(3)^3-3(3)^2-22.5(3)+19\\f(3)=2(27)-3(9)-67.5+19\\f(3)=54-27-48.5\\f(3)=-21.5<0` - Increasing:
`[-\infty,-1.5)(2.5,\infty]` - Decreasing:
`(-1.5,2.5)`

E.
`f(x)=2x^3-3x^2-22.5x+19\\f'(x)=6x^2-6x-22.5\\f'(x)=6x^2-6x-22.5=0\\4x^2-4x-15=0\\(2x+3)(2x-5)=0\\2x+3=0~|~2x-5=0\\2x=-3~|~2x=5\\x=-1.5~|~x=2.5\\x=-1.5,2.5\\\\f''(x)=12x-6\\12x-6=0\\12x=6\\x=0.5` - Concave up:
`(0.5,\infty]` - Concave down:
`[-\infty,0.5)`