## Critical Point Mathematics Questions and Answers

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For the following problem, use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. f(x) = 3x^5-5x^3

For the following problem, use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. f(x) = x^5-5x^4+35

For the following problem, use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. f(x) = 3x^4-4x^

For the following problem, use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither. f(x) = 2x^3+3x^2-3

A function and value of x so that f'(x) =0 are given. Use the second derivative test to determine whether each point (x, f(x)) is a local maximum, a local minimum or neither. h(x)=x \ln(x), \ x=\frac {1}{e}

A function and value of x so that f'(x) =0 are given. Use the second derivative test to determine whether each point (x, f(x)) is a local maximum, a local minimum or neither. h(x)=x^4-8x^2-2, \ x=-2,0,2

Use information form the derivative of the function to help graph the function. Find all local maxima and minima of the function: g(x)=2x^3-15x^2+6

Find all critical points and global extreme of the given function on the given interval: f(x)= x^3-3x+5 on -2,1.

Find all critical points and local extreme of the given function on the given interval: f(x)= x-e^x in the entire real number line.

Find all critical points and local extreme of the given function on the given interval: f(x)=x^3-3x+5 in the entire real number line.

For all of the critical points and local maximums and minimums of the following function: f(x)=x^2+8x+7

Find the critical values of f(x) = 2x^3+3x^2-12x+10

Using Curve Sketching methods, sketch the graph of the function y = x^{3}e^{x}. Make sure that you include all steps, charts, and derivations details.

a. Show that the continuous scalar function of two variables given by f(x,y) = - (x^2 - 1)^2 - (x^2y - x - 1)^2 has two local maxima and no local minima. b. Create a graph of f(x,y) using Mathematica or some other tool. Confirm visually that there are two

Let c be the cut point of f (x) (critical point of f(x)) such that f (c) =0 and f (x) exist for all x then: a. Function f(x) has a relative maximum at the cut point c if f (c) 0, b. Function f(x) has a relative minimum at the cut point c if f (c) 0, c.

Find the critical numbers of the function y = 3x^4 + 8x^3

List the critical numbers of the function f(z) = \frac{2z+2}{7z^2 + 7z + 7} in increasing order

Use Mathematica to find the coordinates of the maximum that occurs for f(x) between x = -10 and x = -5. Write down those coordinates directly on your graph, accurate to 3 decimal places in both x and y. There are many different ways to get Mathematica to

Find the points on the surface 2x^2 + y^2 = (z - 2)^ 2 that are a) closest to the origin and b) furthest from the origin.

Find the maximum and minimum values of f(x, y) = e^(x/y) - y/x + (x - 2)^2 on the trapezoid D = -x less than or equal to y less than or equal to x, 1 less than or equal to x less than or equal to 3.

Consider the function f(x,y) = -2x^2 + 2xy -y^2 +18x - 14y a. Find the stationary point of f(x,y) b. Prove that the stationary point you found in (a) is the global maximum of f(x,y)

Sketch the graph of the given function by determining the appropriate information and points from the first and second derivatives. y = x^3 - 3x^2 - 105x.

1.) Find the increasing and decreasing interval(s) and identify all relative extrema. j(x) = 5-x/x^2-16 2.) Find the interval(s) of concavity and point(s) of inflection. k(x) = x/(2x-5)^5

Find the maximum value of M of the function \\ x^9y^9(6-x-y)^6 \\ on the region x\geq 0, \ y\geq 0, \ x+y \leq 6.

How many critical points can f have? List all possible cases, and provide an example (with explanation) for each case.

Determine the coordinates (x, y) of transition points for f(x) = -x^3 + 2x^2 - 3. Then sketch the graph by determining the sign combination (++, + -, --, or - +) on the appropriate intervals.

A) Find all relative extrema of f(x, y) = 2x^2 - 12x + y^2 + 8y + 34. B) Find all relative extrema of f(x, y) = e^(xy). C) Find all absolute extrema of f(x, y) = x^2 + y^2 on the rectangle 0 less than

For each of the following functions, find the maximum and minimum values of the functions on the circular disk: x^2 + y^2 less than or equal to 1. Do this by looking at the level curves and gradients.

Let f(x,y) = x^2 + y^2 - 6x - 16y + 122 for all (x,y) satisfying the inequalities y \leq 3 + 2x, y \leq 15 - 2x, and (\frac{13}{3})+(\frac{2}{3}) \leq xy. Find the maximum and minimum values on

For each of the following, find the maximum and minimum values of the function on the circular disk x² + y² ≤ 1. Do this by looking at the curves and gradients. A) f(x, y) = x + y + 4:

Use f(c), f'(c), and f"(c) to sketch and describe the graph of (x) = 3^xx^3 at x= -2.

A one-product company finds that its profit P, in millions of dollars, is given by the following equation where ''a'' is the amount spent on advertising in millions of dollars, and ''p'' is the price charged per item of the product, in dollars.
P(a,p)=2a

Find the absolute extrema of the function over the region R. f(x,y)=2xy-x-y, R:\left \{ (x,y) \mid y\leq 4,y\geq x^{2}\right \}

Find the point on the plane z = 13 - 2x - 3y closest to (1, -1, 0). Then prove that it minimizes distance using the second derivative test.

absolute extrema on open and/or unbounded sets ........ Find the point on the surface f(x,y)=x^2+y^2+10 nearest the plane x +2y - z = 0. identify the point on the plane

Find the absolute extrema of f(x, y) = (4x - x^2) cos (y) on the rectangular plate 1 less than equal to x less than equal to 3, - \frac{\pi}{4} less than equal to y less than equal to \frac{\pi}{4}

Find the absolute extrema of f(x,y) - 4x^2 + 2y^2 + 24x - 8 on the region R = \left \{ {(x, y):x^2 y^2 \leq 49} \right \}

Consider the function f (x,y )=xy-5y-25 x+125 on the region on or above y=x^2 and on or below y=27. (a) Find the absolute minimum value. (b) Find the points at which the absolute minimum value is attained. (c) Find the absolute maximum value. (d) F

Find the absolute maxima and minima of the following functions on the given region R. a) f(x,y)=2x^{2}+y^{2}\ on\ R=\left\{(x,y):\ x^{2}+y^{2} is less than or equal to 16\right\} b) f(x,y)=6-x^{2}-4y^

Find the absolute extreme of the function over the region R. f(x, y) = 3x^2 + 2y^2 - 4y, R: The region in the xy-plane bounded by the graphs of y = x^2 and y = 4.

Find the critical points of the differential equations using the existence and uniqeuness theorem. a) \frac{dy}{dx} = y^2 - 3y b) \frac{dy}{dx} = y^2 - y^3

Find the absolute extrema of the function f(x,y) = x^2 + y^2 - 2y + 1 in the closed region bounded by the curves y = x^2 and y = x + 2.

Suppose that f(x, y) = x^2 - xy + y^2 - 4x + 4y with D = {(x, y) | 0 \leq y \leq x \leq 4}. 1. The critical point of f(x, y) restricted to the boundary of D, not at a comer point, is at (a, b). Then

Find the maximum and minimum distance from graph x^3 + y^3 = 1 , x \geq 0 , y \geq 0 to points (-1, -1).

Let f(x,y)=x^2+y^2-8x-12y+82 for all (x,y) which satisfy the inequalities y less than or equal to -1+2x , y less than or equal to15-2x and 5/3+2/3x less than or equal to y . Find the maximum and mini

Find the exact extreme values of the function z = f(x, y) = 33x + 11y + 60 subject to the following constraints: y greater than or equal to 0, y less than or equal to 9 - x^2.

Find the extreme values of f (x, y, z) = x^2 y z + 1 on the intersection of the plane z = 3 with the sphere x^2 + y^2 + z^2 = 15. (a) The minimum of f (x, y, z) on the domain. (b) The maximum of f (x,

Find the absolute minimum of f over __D__. f(x, y) = x^4 + 2y^4 ? 4yx^2 + 10 over the region D = {(x, y) \epsilon \ R 2 \ |x ≥ -2, y ≥ 0}. consider the points on the boundary too.

State whether the following statement is true or false. If it is false, correct the statement and explain why it is not true, or give a counter example. If f'(c) = 0 , then f(x) has a local maximum or minimum at c

Domain: ( \infty, \infty) Derivative =0 at x=1 Derivative undefined at x= -3 and x=4 Test numbers: f'(0)= 12 \\ f'(4)=28 \\ f'(10)=20 \\ f'(1.25)= 5,4321 a. What are the critical numbers if any? b. Where is the function increasing? c. Where is the func

Consider the function f(x) = 9 - |x - 3 | a. Find the critical numbers of the function. b. Find the open intervals where the function is increasing or decreasing. c. Apply the First Derivative Test to identify the relative extremum.

Find all critical points of the function f(x) = 36x^2 \sqrt{64 - x^2}.

Find the positive critical point of the function f(x) = \frac{x}{x^2 + 3}.

Sketch the graph for a single function that has all of the properties listed. f'(-4)=f'(3)=0

Consider the function f(x)= x^2e^{-x}. Find the critical points of f.

Find the value of x that minimizes y= 7x^2+\frac{1900}{x} for positive x.

Find the exact coordinates of the inflection points and critical points of the function f(x) = 4x^5 - 100x^3 - 5

Find the exact coordinates of the inflection points and critical points of the function f(x) = \frac{2}{3}x^3 -2x^2 - 6x

Use the First Derivative Test to find and classify each of the critical points of the function f(x) = -\cos x - x \sin x in the open interval (-3,3)

Let g(x,y) = xye^{ x^2 y^2} . 1. Find all critical points. 2. Classify the critical points found above.