Answered

Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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

Sagot :

Since3122015
A. [tex]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[/tex] - Relative maximum at x = -1.5. Relative minimum at x = 2.5. Point of inflection at x = 0.5.

B. [tex]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[/tex] - Absolute maximum at x = -1.5. Absolute minimum at x = 2.5. This applies for the interval [-4, 5].

C. [tex]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[/tex] - Absolute maximum at x = 0. Absolute minimum at x = 2.5. This applies for the interval [0, 3].

D. [tex]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[/tex] - Test: [tex]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[/tex] - Increasing: [tex][-\infty,-1.5)(2.5,\infty][/tex] - Decreasing: [tex](-1.5,2.5)[/tex]

E. [tex]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[/tex] - Concave up: [tex](0.5,\infty][/tex] - Concave down: [tex][-\infty,0.5)[/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.