Answer to Question #263705 in Calculus for Alunsina

Question #263705

In numbers 1-4,

(a) find the critical numbers of the given functions

(b) using either the First or Second Derivative Test, classify whether thd function has a relative maximum or minimum at these critical numbers

(c) find the coordinates of the exterum points

(d) find the points of jnflection

(e) find the x and y - intercepts

(f) summarize the points in a table. include additional points necessary for the graph

(g) sketch the polunomial curves in graphic papers


1. f(x)=x³+x²-x+2

2. f(x)=(x4/4)-(x³/3)-2x²+4x+3

3. f(x)=x4-8x²+9

4. f(x)=x³+6x²+9x+3


1
Expert's answer
2021-11-18T06:53:48-0500

1(a) Critical points of "f(x)=x\u00b3+x\u00b2-x+2"

steps

"find\\ f'(x)=3x^2+2x-1"

find X

"Critical \\ points\\ x=-1, x=\\frac{1}{3}"



1(b)"f(x)=x\u00b3+x\u00b2-x+2"

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

"f'(x)\\gt 0:x\\lt -1 \\ or \\ x\\gt\\ \\frac{1}{3}"


"f'(x)\\lt0:-1\\lt x\\lt \\frac{1}{3}"




1(c) Coordinates of extrema points "of \\ f(x)=x\u00b3+x\u00b2-x+2"

Local maxima at "x=-1"

"coordinates \\ = (-1,3)"


local minima at "x=\\frac{1}{3}"

"=(\\frac{1}{3},\\frac{49}{27})"


1(d)Points of inflection

"f"(x)=6x+2\\\\find \\ where \\ f"(x)=0 \\ or \\ undefined: \\ x=-\\frac{1}{3}"

"plug \\ x=-\\frac{1}{3} \\ into \\ f(x)=x\u00b3+x\u00b2-x+2"


"=(-\\frac{1}{3},\\frac{65}{27} )"


1(e)

"for \\ x \\ intercepts \\ replace\\ y\\ with\\ 0\\\\ for \\ y \\ intercepts \\ replace\\ x\\ with\\ 0"

"\\mathrm{X\\:Intercepts}:\\:\\left(-2,\\:0\\right),\\: \\\\ \\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:2\\right)"


1(f) plot the points "for f(x)=x\u00b3+x\u00b2-x+2"




1(g) Graph "f(x)=x\u00b3+x\u00b2-x+2"







2(a) "critical \\ points\\ f\\left(x\\right)=\\left(\\frac{x^4}{4}\\right)-\\left(\\frac{x^3}{3}\\right)-2x\u00b2+4x+3"

"find\\ f'(x)=4x^3+3x^2-4x+4\\\\solve\\ for\\ x"

"x=-2\\\\x=1\\\\x=2"


2(b) first Derivative test for maxima and minimum

"f'(x) = 4x^3-3x^2-4x+4"


"f'(x)\\gt0: -2\\lt x\\lt1\\ or \\ x\\gt2"

"f'(x)\\lt0: x\\lt-2\\ or \\ 1\\lt x\\lt2"





2(c) Coordinates of extrema points

"local\\ maxima = (1, \\frac{59}{12})\\\\local\\ minima=(-2,-\\frac{19}{3} )\\\\local minima =(2,\\frac{13}{3})"


2(d) points of inflection of "f\\left(x\\right)=\\left(\\frac{x^4}{4}\\right)-\\left(\\frac{x^3}{3}\\right)-2x\u00b2+4x+3"

"f"(x)=3x^2-2x-4\\\\find \\ where \\ f"(x)=0 \\ or \\ undefined: \\ x=\\frac{1-\\sqrt{13}}{3}, x=\\frac{1+\\sqrt{13}}{3}"


"plug \\ x=x=\\frac{1+\\sqrt{13}}{3} \\ and \\ x=\\frac{1-\\sqrt{13}}{3}\\ into \\ f(x)=\\frac{x^4}{4}-\\frac{x^3}{3}-2x^2+4x+3"

"=\\left(\\frac{1-\\sqrt{13}}{3},\\:-\\frac{2\\left(61+35\\sqrt{13}\\right)}{81}+3\\right),\\:\\left(\\frac{1+\\sqrt{13}}{3},\\:\\frac{2\\left(35\\sqrt{13}-61\\right)}{81}+3\\right)"


2(e)

"for \\ x \\ intercepts \\ replace\\ y\\ with\\ 0\\\\ for \\ y \\ intercepts \\ replace\\ x\\ with\\ 0"

"X \\ intercepts=\\left(-0.59732\\dots ,\\:0\\right),\\:\\left(-2.87680\\dots ,\\:0\\right)\n\\\\\\:\\mathrm{Y\\:Intercepts}=\\:\\left(0,\\:3\\right)"


2(f) plot the points"f\\left(x\\right)=\\left(\\frac{x^4}{4}\\right)-\\left(\\frac{x^3}{3}\\right)-2x\u00b2+4x+3"




2(g) graph





3(a)Critical points "of \\ f(x)=x^4-8x^2+9"

"find\\ f'(x)=4x^3-16x\\\\solve\\ for\\ x"

"x=-2,\\:x=0,\\:x=2"


3(b) using first derivative test

"f'(x)= 4x^3-16x\\\\f'(x)\\gt0: -2\\lt x\\lt0 \\ or \\ x\\gt2\\\\f'(x)\\lt0: x\\lt-2 \\ or \\ 0\\lt x\\lt2"





3(c) coordinates of extrema points

"local\\ maxima = (0, 9)\\\\local\\ minima=(-2,-7 )\\\\local\\ minima =(2,7)"



3(d) points of inflection "of \\ f(x)=x^4-8x^2+9"

"f"(x)=12x^2-16\\\\find \\ where \\ f"(x)=0 \\ or \\ undefined: \\ x=-\\frac{2\\sqrt{3}}{3}, x=\\frac{2\\sqrt{3}}{3}"


"plug \\ x=-2\\ into \\ f(x)=x^4-8x^2+9=\\frac{1}{9}"


"=\\quad \\left(-\\frac{2\\sqrt{3}}{3},\\:\\frac{1}{9}\\right),\\left(\\frac{2\\sqrt{3}}{3},\\:\\frac{1}{9}\\right)"


3(e)

"for \\ x \\ intercepts \\ replace\\ y\\ with\\ 0\\\\ for \\ y \\ intercepts \\ replace\\ x\\ with\\ 0"

"\\mathrm{X\\:Intercepts}:\\:\\left(\\sqrt{4+\\sqrt{7}},\\:0\\right),\\:\\left(-\\sqrt{4+\\sqrt{7}},\\:0\\right),\\:\\left(\\sqrt{4-\\sqrt{7}},\\:0\\right),\\:\\left(-\\sqrt{4-\\sqrt{7}},\\:0\\right),\\\\ \\:\\mathrm{ Y\\:Intercepts}:\\:\\left(0,\\:9\\right)"



3(f) Plot the points "of \\ f(x)=x^4-8x^2+9"





3(g) graph




4(a) critical points of f(x)=x³+6x²+9x+3

"find\\ f'(x)=3x^2+6x+9\\\\solve\\ for\\ x"


"x=-3,\\:x=-1"



4(b) "f'(x)=3x^2+12x^2+9"


"f'(x)\\gt0 : x\\lt-3\\ or\\ x\\gt-1\\\\f'(x)\\lt0: x\\lt x\\lt-1"


4(c) coordinates of Extrema points of f(x)=x³+6x²+9x+3

"local\\ minima = (-1, -1)\\\\local\\ maxima=(-3,-3)"


4(d) Points of inflection of f(x)=x³+6x²+9x+3

"f"(x)=6x+12\\\\find \\ where \\ f"(x)=0 \\ or \\ undefined: \\ x=-2"


"plug \\ x=-2\\ into \\ f(x)=x\u00b3+6x\u00b2+9x+3=1"


"=\\left(-2,\\:1\\right)"



4(e)

"for \\ x \\ intercepts \\ replace\\ y\\ with\\ 0\\\\ for \\ y \\ intercepts \\ replace\\ x\\ with\\ 0"

"\\mathrm{X\\:Intercepts}:\\:\\left(-0.46791\\dots ,\\:0\\right),\\:\\left(-1.65270\\dots ,\\:0\\right),\\:\\left(-3.87938\\dots ,\\:0\\right),\\:\\\\ \\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:3\\right)"



4(f) plot the points of f(x)=x³+6x²+9x+3




4(g) Graph f(x)=x³+6x²+9x+3






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