 #1
gassi
 3
 0
I have two problems. I posted the first problem before but I still can´t solve it.
Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3

df/dx = y + z +3x^2, df/dy = x + z + 3y^2 and df/dz = x + y + 3z^2
a point x is a critacal point if the gradient equals 0.
Obviously (0,0,0) is a critical point but I´m not sure how to find the others.
I know this is symmetric but I cant´t figure out were to go from here??
Here is the second problem:
I have a function from R to R, f(x) = x + 2*x^2*sin(1/x) if x is not 0 and f(x) = 0 if x is 0.
I´m supposed to show that this is differentable everywhere, that f'(x) is not 0 and that f maps no neighbourhood around the zero point bijective on a neighbourhood around the zero point.
I think I know how to show that this is differentable everywhere, that f'(x) is not 0 but I´m having difficulties with the last one.
Homework Statement
Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3
Homework Equations

The Attempt at a Solution
df/dx = y + z +3x^2, df/dy = x + z + 3y^2 and df/dz = x + y + 3z^2
a point x is a critacal point if the gradient equals 0.
Obviously (0,0,0) is a critical point but I´m not sure how to find the others.
I know this is symmetric but I cant´t figure out were to go from here??
Here is the second problem:
I have a function from R to R, f(x) = x + 2*x^2*sin(1/x) if x is not 0 and f(x) = 0 if x is 0.
I´m supposed to show that this is differentable everywhere, that f'(x) is not 0 and that f maps no neighbourhood around the zero point bijective on a neighbourhood around the zero point.
I think I know how to show that this is differentable everywhere, that f'(x) is not 0 but I´m having difficulties with the last one.