Answer to Question #199231 in Calculus for Nikhil Rawat

Question #199231

Let f: R^3→R be function defined by :

F(x,y,z)= |x+2y+z| . Show that f is not differentiable at the point (1,-1,1).


1
Expert's answer
2021-06-01T09:10:21-0400

If we take example of "|x|" 

Then, it is clearly not differentiable at "x=0" but if we choose some other value then the given function is differentiable .

So, the function |"x|" is continuous at "x=0" but not differentiable at "x = 0" .




but the given function in the problem is 3D Function and if we see the given points which is

 "R^{3}\\to {R}" "f(x,y,z) =" |"x+2y+z|" 

and if we put this point where we have to prove that given function is differentiable or not then the given "f(x,y,z)=0" which shows that at this given point clearly the function is not differentiable.


So, the function "f(x,y,z) =" "|x+2y+z|" is continuous at the point (1,-1,1) but not differentiable at that given point so from the given example of |x| we prove for this given function

"f(x,y,z)" "=|x+2y+z|" and the function is differentiable at all the other point rather than (1,-1,1) which is given in the question.

And if we choose any other point rather than (1,-1,1), the the given function is continuous and differentiable both at that point.




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