f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an one-to-one function?

Q: (a) Assume that g(x, y) is a function of the form x²+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that…

A: (a) The given function is gx,y=x3yx2+y2  if x,y≠0,00             if x,y=0,0. To show: (i) gx,y≤xy…

Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

Q: Find the interval that the fullowing function is continous at In (9 -x) %3D A. [I,3] BL) (3,3) (1,3)…

A:

Q: → [0, ∞) defined by f (x) = Vx + 2. Prove that f is 17. Consider the function f: (-2, 0) surjective.

A:

Q: Prove that f: R R where f (x) = |x| is neither injective surjective. nor

A: To prove this theorem, we use the defination of one one and onto.

Q: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c…

A: The Intermediate Value Theorem that states if a function f(x) is continuous and k is     any…

Q: f(x, y) = x² + y² +1 f (x, y)

A: Let's find.

Q: Determine whether the function f: R- R, f(x) =,x # 3 is surjective. x+1 %3D r-3

A: Consider the provided question, We need to determine whether the function f:R→R fx=x+1x-3, x≠3 is…

Q: Let f be analytic in a region G. If f(g) lies on the parabola u=v^2, prove that f is constant in G.

A:

Q: Consider the functionf defined by f(x,y) = k. Use the definition to show that for fixed real numbers…

A: We are given that f(x, y) = k and a, b, c, and d are fixed real numbers with a<b and c<d.…

Q: Let f : R² → R with f(x,y) = x²+ y. Prove that f is differentiable at the point (0, 0)

A: Given function is f(x,y)=x2+y Since we know that, differentiability of function of two variable at…

Q: Prove whether u(x, y) = x² - 2xy + y2 is a harmonic function on a set D={(x,y) |x<y}. If and are…

A: What is Harmonic Function: A function of multiple variable is referred to as harmonic when its…

Q: Prove that af +y. -yof=nf (x, y). əx (Hint: Differentiate both sides of the definition of a…

A: Given  a) x∂f∂x+y∂f∂y=nf(x,y)b) x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=n(n-1)f(x,y)

Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: A function, F: R × R ---› R x R has been defined as follows: F(x, y) = (3y - 1, 1 - x) for all (x,…

A:

Q: Let f be differentiable on the closed interval [a, b] such that f′(a) = y1 and f′(b) = y2. If d lies…

A:

Q: Determine whether the function f(z) = e is analytic or not using the Cauchy- Riemann theorem.

A: In the given question we have to check the analiticity of f(z)=ez2 by using the C.Requation.the C.R…

Q: Let f : R² → R with f(x, y) = Vx² + y². Sketch the graph of f. Prove that f is not differentiable at…

A: Consider the given function, fx,y=x2+y2 Now, draw the graph of the above function.

Q: • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b).…

A: We have to give an example of a function f(x) that is not bounded on [a, b].

Q: Find and sketch the domain of f(r, y) In(r - y) 6. -

A: Given:      f(x,y)=lnx-y9-x2-y2·x2+y2-4    Interval notation : -3,0∪0,3   Set builder notation:…

Q: Show, using the definition of continuity from the topological point of view, that the constant…

A: See the attachment.

Q: Use Euler’s Theorem to find the degree of homogeneity for f( x1 , x2) = x3 + y3 + 3xy2

A: We will use the Euler's theorem which is : x•fx + y•fy = nf  Where, n = degree of homogeneity    The…

Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

Q: Let f(x) =esin 3x + 2x a. find the linearization of f(x)=esin3x+2x at x=0. b. Use your linearizato…

A: a). Given f(x) =esin 3x + 2x f(x) ≈ L(x) = f(a)+f′(a)(x−a). where the function L(x) is called the…

Q: Let f(2) = z2 + 2|2|2 + z where f is differentiable. Determine the set of all points zo where f is…

A:

Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: Let f(x) be twice differentiable and invertible function such that f'(x) >0 and f"(x) > 0, prove х,…

A:

Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: Q3/ 1- Let f: Z, → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your %3D answer.

A:

Q: Show that there does not exist a function f(x, y) such that VJ= (y2,x). Hint: Use Clairaut's Theorem…

A:

Q: Apply the theorem of Cauchy-Riemann equations to verify that function f (z) is entire: f (z) = 3x +…

A:

Q: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c…

A: Intemediate value theorem gives that if f(x) is a continuous finction within the interval a, b  then…

Q: Without the aid of Wrongskian, determine whether the given set of functions is linearly independent…

A:

Q: Find the domain of the function f(x,y)=V64613-x + y-x² ° O {(x,y) e R2 | y# x²}. О {({х,у) € R2 |…

A: Given the function

Q: Let f (x) = vr +2² + 2 , ro = 1, h = 0.5, r1 using the Three-Point formulas to approximate f(1) (1);…

A: Given: The given fx=x3+x2+2, x0=1,h=0.5,x1=1.5 and x2=2

Q: Verify that Rolle's Theorem can be applied to the function f (z) = 23 – 10z2 +27x – 18 on the…

A:

Q: Consider the function f: (-2,0) → [0, 0) defined by f (x) = Vx + 2. Prove that f is suriective.

A:

Q: 4|x| – 3 pour x 1, x² – 4 f2(x) + V2 + x², || ||

A: Given that, f1x=4x-3x≤1-3xx2-4x&gt;1f2x=1+2+x2 To determine set of points which they are derivable…

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = (x, y) = (0,0) y+2xy² x²+y2 Let…

A:

Q: Let g(u) be a differentiable function and let f(x, y) = g(x² + y²), find the direction of maximal…

A:

Q: Let f : R → R be such that for all x, y ∈ R, |f(x) - f(y)| ≤ (x - y)2. Prove that f is a constant…

A:

Q: Prove that the extreme value theorem doesn’t work over ℚ. That is, find a continuous function, f: ℚ…

A: Consider the function f: ℚ→ℚ, where f(x)=sin(x). Consider the function in the closed interval 0, 2…

Q: Let f be the function defined by f(2) = x³ + iy³ . Determine the points z E C where f is complex…

A:

Q: ) Use Stoke's theorem to evaluate f f.dr in the counterclockwise direction when viewed from the…

A:

Q: Let f : [0, ∞) → [0, ∞) be the function defined by f(x) = √x. Show that f is a bijection.

A:

Q: Discuss the continuity in R3 of the function f(x, y, z) %3D x2+ y2+z2 -1

A: Given: f(x,y,z)=1x2+y2+z2-1.

Q: Let fi and f2 be concave functions from R" to R. Let f : R" +R be defined as the pointwise minimum…

A:

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

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.