(a) Let f: R → R be a differentiable function such f(tx) = tf(x) for all t ≤ R, x € Rª and some fixed me N. Show that: n [ i=1 af Xi ¹Jxi = mf(x) = Hint: Consider the function g(t) = f(tx) and g'(1) (b) Let f: Rn → R be a non-linear function such that f(tx) = tf(x) for all t = R, x ER". Show that f is not differentiable at 0.

Q: Find all differentiable functions f: R→R such that f'(x) = F(x + n) – f(x) n for all real numbers x…

A: As f'(x)=f(x+n)-f(x)n is true for all positive integers values of n thus substitute 1 for n in…

Q: Consider the following function f: R -> R,whose criterion is f(z) = 43x' + 6x – 5 + e9(z) with g a…

A: If you like the solution then please give it a thumbs up.. The Answer is:        f''(a) = 258a + 81…

Q: 1. Let h : R → R be some twice-differentiable function. Define f(x,y) =h() for all x ≠ 0. Show that…

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Q: Use the second derivative f"(x) = x(x+ 1)(x-3) to determine where the function f(x) is concave down.…

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Q: Find the domain of f(z) the analytic function, graph on the z-plane.find the Re(f'(1)). f(z) = (z² +…

A: Given is the function f(z)=z2+1i. We need to find the domain of the function.

Q: Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0}…

A: To show that the function f defined as below is differentiable on A and compute its derivative

Q: Let f(2) = x* – i(y + 1)°, (z = x + iy), (a) Determine the points where f is C-differentiable, and…

A: Given :-  let f z = x4 - i  y + 13 where z = x + iy (a) The points where f is ℂ - differentiable ,…

Q: Let f: A-XX Y be defined as f (a) = (f, (a). f2 (a)) each a A, Then f is continuous if and only if…

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Q: Suppose f is a differentiable function of x and y, and g(r, s) = f(2r - s, s² - 7r). Use the table…

A: ​gr​(7,6)=−50gs​(7,6)=93​Explanation:Step 1:​Given a Table of values of f,g and fx​,fy​​​To find:…

Q: If a and b are real numbers and f(x) and g(x) are differentiable functions such that fla)=g(a)=1,…

A: Introduction: The by parts method is a method to determine integration. When the integrand is given…

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: The function f(x, y) = ¿e=y – In(y²)

A: It is partial derivative

Q: sider the function f (t) = –2t³ – 6t² + 8 on the interval [-1, 2]. of values of c for which such…

A: f'(c) = f(b) -f(a) /b-a

Q: Determine if the following functions are analytic or not: 1. f(z) = (z* + 1)³ – 3z* where z is the…

A: (i) The  given function is ;               fz = z*+13 - 3z* We know that a function  f is analytic…

Q: 2. Numerical Methods. Using h = 0.2, approximate the value of y (3) for the given differential…

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Q: Let fbe adifferentiable function satisfying the condition. f(x) for all x, y (+0) ERand f(y)#0. f(y)…

A: Let f be differentiable function satisfying the condition Given f(xy)=f(x)f(y)put…

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Q: Being fa differentiable function. Consider the relation where z is implicitly defined in terms of x…

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Q: Find a multivariable function f(x, t) that satisfies the Black-Scholes equation: ft = f - xfx -…

A: f(x,t)=etlog(x) satisfies Black-Scholes equation: ft = f - xfx - x2fxx

Q: Let I be and interval and let f : I →R and g : I → R be functions differentiable at c. If h : I → R…

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Q: Construct an analytic function of the form f (z) = u + iv, where v is tan- (y/x), x # 0, y # 0

A: f(z)=u+iv where v=tan-1(yx)  We need to find the analytic function f(z)=u+iv  we apply Milne Thomson…