Answer to Question #106745 in Calculus for Nikesh gautam pandit ji

Question #106745
State whether the following statements are true or false. Give reasons for your answers.
a) The function R^3 → R. f : , given by f (x, y,z) = |x| + |y| + |z| is differentiable at
3,2(-1)

b) The function
f(x,y)=max{y/x,x} is a homogeneous function on R^2

c) The domain of the function f / g where f (x, y) = 2xy and g(x, y) = x^2 + y^2 is R^2 .
1
Expert's answer
2020-03-30T04:38:27-0400

a) True

The given function

"f:\\R^3\\rightarrow \\R" is defined by "f(x,y,z)=|x|+|y|+|z|."

Consider,the following maps.

"f_1:\\R^3\\rightarrow \\R" defined by "f_1(x,y,z)=|x|" ,

"f_2:\\R^3\\rightarrow \\R" defined by "f_2(x,y,z)=|y|" ,

"f_3:\\R^3\\rightarrow \\R" defined by "f_3(x,y,z)=|z|" .

Clearly,"f_1,f_2,f_3" are differentiable at "(3,2,-1)" ,since the absolute value functions are differentiable everywhere except "(0,0,0)".

Again, we known that sum of differentiable functions is again differentiable.

Therefore,"f:\\R^3\\rightarrow \\R" defined by

"f(x,y,z)=f_1(x,y,z)+f_2(x,y,z)+f_3(x,y,z)=|x|+|y|+|z|"

is differentiable at "(3,2,-1)" .


b) False.

A homogeneous real valued function of two variable x and y is a real valued function that satisfies the condition

"f(rx,ry)=r^kf(x,y)" for some constant "k" and all real number "r."

Now, the given real valued function of two variable is

"f(x,y)=max\\{ \\frac{y}{x},x \\}" .

Putting,"x=rx \\ and \\ y=ry" ,we get

"f(rx,ry)=max(\\frac{ry}{rx},rx)=max(\\frac{y}{x},rx)\\neq r^kf(x,y)."

for some k and all real number r.

Hence,"f" is not a homogeneous function.



c) False.

Given function are

"f:\\R^2:\\rightarrow \\ \\R" defined by "f(x,y)=2xy."

"g:\\R^2:\\rightarrow\\R" defined by "g(x,y)=x^2+y^2" .

Now ,

"\\frac{f}{g}(x,y)=\\frac{2xy}{x^2+y^2}"

but at "(0,0),"

"\\frac{f}{g}(0,0)=\\frac{0}{0}"


which is not defined.

Hence , "\\R^2" is not a domain of "\\frac{f}{g}" .


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