f(b)-f(a) = Gradient(f(c)) DOT (b-a). Are my steps right? how do i solve the final equation i arrive at?

Question

ok, i am just making sure this is correct. I am givin this.

f(x,y,z) = x^2 - y*z +z^ 2
a(0,1,1) b(1,3,2)

find a point c on the line joining a and b such that:
f(b)-f(a) = Gradient(f(c)) DOT (b-a).

Heres my work:

f(b) - f(a) = 2
b - a = (1,2,1)
f(c) = xo^2 - yo*zo +zo^2
Gradient(f(c)) = (2xo*I - zo*J + (2zo - yo)*K)

after i figured all that out i straight subsituted into the original proof statement:
f(b)-f(a) = Gradient(f(c)) DOT (b-a).

After simplifying i got:

2xo - yo - 2 = 0

why did my zo dissapear (if it didnt then what?) and can i just pick any number for xo or yo and solve for the other then my point will be:

c (1,0,0) if i set xo to 1

Please verify or falsify and help. Thanks in advance.

Answer 1

I think that you need to use the chain rule on that second term of your f(x,y,z)