5. Find the domain and range of the following functions algebraically: (a) y = 1+ v7x – 2 (b) y = (x + 4)(x – 2)

Q: I) Consider the function: y = + C,e2x + C,e-5x + Czxe-5x

A:

Q: Find the values of x for which  f(x)=(1/3)x3 +(5/2)x2 +6x-17 Has a tangent line that is parallel…

A: As the line is parallel, slope of them will be equal. Slope of f(x)=x33+5x22+6x-17 will be :…

Q: Determine equations of all lines tangent to the graph of f(t) = t° /3+2t² +4t that have slopes of 1.

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Q: B- Find the local extreme points of the function f(x, y) = 4xy - x - y^.

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Q: The function y=f(x) satisfies the conditions y(0) = 1 and y'=xy³/4. What is y(1)?

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Q: Find the domain of function defined as follows. y = ln(x2 - 16)

A: y=lnx2-16Domain:       The domain of a function is the set of all possibleinputs for the function

Q: 2 Z= 22,

A: Given  z=2x12+x1x2+4x22+x1x3+x32+2

Q: If the critical point of the function f(x.y)-x- y- 2xy +6 is (0.0). test this point whether it is:

A: We use second order partial derivative to know about critical point behaviour

Q: B. Find and classify the extreme points of function f(x, y) x* +y-4xy-1.

A: Maxima minima of two variables function

Q: Are the f-llowing functions Continual s OFC メーE X-2 スト=メ 1P 2) FCX)s X- S X +7 at XE-1 X5qXs (xフゴ

A:

Q: Find the the stationary points and the local extreme values of the function f(x, y) = −2x/x2 + y2+1

A: The objective is to find the stationary points and the local extreme values for the function…

Q: Find f'(2) if y = f(x) is the nonnegative function defined for x, y > 0 by the equation x" = y?.

A: To find f’ (2), if y = f (x).

Q: Find all critical points of the function y= x^4 - 8x^2.

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Q: HW 1-Find the domain for each function : (1) f(x,y)=Vy-x-2 (2) f(x,y)= In(x²+y² -4)

A: We have to find the domain from the given functions

Q: 3) Let f (x) = e=x² Where is y'> 0? What does this tell you about the original function?

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Q: 1-Sketch the curve for f(x)= =-2r+4 x-2

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Q: Find the local and absolute extremum for the functions over their domain: 5. f(x) = vx2 + 8x + 20

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Q: how would you find the x-values that the function f(x) = 2e^x - 6x have horizontal tangent lines? I…

A: The given function fx=2ex-6x

Q: Find all critical points of the function:  f (t) = 8t3 − t2

A: If f has a local extremum (maximum or minimum) at C and if f'(c) exists, then f'(c)=0. A critical…

Q: d) Find the value of the constants ' a’ and b ’ that makes the function continuous at x= 3 ах + b ;…

A:

Q: 1. Come up with the equation of any function f (x, y) satisfying fx = x + y and fy = x + 3y³

A: The objective is to find the function f(x,y) satisfying fx=x+y and fy=x+3y3 Now take, fx=x+y Now…

Q: 10. Find a function y = ax +bx+c whose graph has an x-intercept of 1, a y-intercept of -2, and a…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (b) Write the equation of the line tangent to the graph of f(x) at z 1, and graph it below f(x) =…

A:

Q: Find the critical point of the function f(x, y) = – 7 2x + 5x2 - 8y – 5y? |

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Q: 2. Find the domain and range of the function y = Vx + 4

A: We Know that  Domain: The set of all points over which a function is defined. Range: The set of…

Q: Where is the function y =4x/x-3 concave downward? Show mathematical work to clearly answer this…

A: For a given function y=f(x) the curve is concave downward in an interval I when the second…

Q: 4. a) Find the value of x where the function f(x) = (x – 4)5 is not differentiable.

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Q: Find the streamlines corresponding to: f(z) = z²

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Q: Find dydx, where y is defined as a function of x implicitly by the equation below. y3+2x2y5=1

A:

Q: 5. Create a graph of the function: t -1 %3D x+2 Ay

A: Given:         Equation is y=-1x+2below graph is plotted using online tool

Q: Show that the function f(1) = (e¯1 + 1)¯' satisfies y' + y² = y, y(0) = }.

A: To prove: ft=e-t+1-1 is a solution of given differential equation y'+y2=y.

Q: Q3. At what points are the functions continuous y =-3x x+3 y = x2-3x-10' y = (2x - 1) X-2

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: 4. Find all the points at which the direction of fastest increment of the function f(x, y) = x² + y²…

A: According to question,Given Function,fx, y = x2 + y2 -2x -2y

Q: Determine the stationary values of the function z = x3 – 6x² – 8y² and distinguish between them

A:

Q: Suppose g(-2)=3 and g'(-2)=-4. Let f(x) = x3 + g(x). Find the equation of the line tangent to y =…

A: Given,     f(x)=x3+g(x)

Q: Find the linearization of the function f(x) = 1/(9x+8) at x = -1.

A: Topic : function

Q: find the critical points in the domains of the following functions. y = ( x^2 - 1 ) / (x^2 + 2x -…

A: It is given that the function is y=x2-1x2+2x-3. Obtain the domain of the function as follows:…

Q: 3. a) Use the Chain Rule and the fact that |x| = Vx2 to show that d dx |x| b) Sketch the graphs of…

A: We will use the give solution as we are instructed.

Q: Determine all the critical points for the function y = Vx² (2t- 1)

A: Given function is in two variables x and t For critical points we calculate Partial derivatives…

Q: Suppose y is a differentiable function of r. Find y' if 2xy 3 2y3 + 5x4. y2 -

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Q: B. Find and classify the extreme points of function f(x, y) = x + y - 4xy-1.

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Q: fla)= 9 (x=4)° +2 +2

A: Given fx=9x-42+2.

Q: Find k so that the function f(x) = x² + kx will be tangent to the line y = 2x-9

A: Here, the given function is: f(x)=y=x2+kx...(1) And also given that the tangent line to the given…

Q: 4. a) Find the value of x where the function f(x) = (x – 4)3 is not differentiable.

A: According to power rule of differentiation , .

Q: Find the level curve of the function at c=2 for g(x,y)=(x^3-y)

A: A scalar function g(x,y) is a rule that is used to associate a number or quantity with a particular…

Q: A company has 100 units to spend on equipment and labour combined. company spends x units on…

A: The domain of a function is the set of values for which the function is defined. To find the maximum…

Q: For the function y = (f ¹)'(-4)= f(x) = x² + 4x - 3, x ≥ - 2, find df-1 dy y=-4

A: Introduction: For a function y=fx, its inverse function is represented as x=f-1y. It is not…

Q: 19. Given the following functions: y = 8x – x2 and y = 3x %3D | Find their points of intersection.

A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…

Q: Find the critical point of the function f(x, y) - 4+ 7x – 5x² + 6y + 7y?

A:

Q: Find the line tangent to the function at the indicated point: f(x) = v8x² – 4x ;x = 1 ОУ-12х-10 o…

A: Since there are multiple questions present and you have not specified which question to answer,…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.