Question

Let f(x, y) = x^3 − 4xy^2 , x, y ∈ R. Use the definition of differentiability to show that f(x, y) is differentiable at (2, 1).

Answer #1

Let f : R → R be a function satisfying |f(x) − f(y)| ≤ 3|x −
y|^{1/2} for all x, y ∈ R. Apply E − δ definition to show that f is
uniformly continuous in R.

Find the absolute extrema of the function over the region
R.
f(x,y) =x^2−4xy+2
R= {(x,y): 1≤x≤4, 0≤y≤2}
Show plenty of steps please! If possible, fairly neat please!

Use the definition of the derivative to show that f(x) = |x^2 –
9| is not differentiable at x = 3

Let f(x) = 5x2 + x − 2. Use the limit
definition of f′(a) and the Limit Theorems only to show
that f′(2) exists and equals 21.

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)
= e^(−1/x^2) if x > 0.
Prove that f is indefinitely differentiable on R, and that
f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a
converging power series expansion En=0 to ∞[an*x^n] for x near the
origin. [Note: This problem illustrates an enormous difference
between the notions of real-differentiability and
complex-differentiability.]

Calculate differentiability of f(x,y,z) = x^2 + y^2 + z^2
this function is defined in R^2

Use Lagrange Multiplier to find the maximum and minimum of
f(x,y) = 4x^2 - 4xy +y^2 subject to 25 = x^2 + y^2.

Let x = [1, 1]T , y = [1, 1]T ∈ R 2 and let f : R 2 =⇒ R 2 with
f(z) =z1.x + z2.y for any z = [z1, z2] T ∈ R 2 . Further, z = g(r)
= [r 2 , r3 ] where r ∈ R . Show how chain rule is applied here
giving major steps of the calculation, write down the expression
for ∂f ∂r , and also evaluate ∂f/ ∂r at...

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable.
Then f and g differ by a constant if and only if f ' (x) = g ' (x)
for all x ∈ [a, b].
b) For c > 0, prove that the following equation does not have
two solutions. x3− 3x + c = 0, 0 < x < 1
c) Let f : [a, b] → R be a differentiable function...

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