Chapter 13.5, Problem 68ES

Prove. If $f,f_x,\text{and}{f}_y$ are continuous on a circular region containing $A\left(x_0,y_0\right)\text{and}B\left(x_1\text{,}{y}_1\right),$ then there is a point $\left(x^{\ast },y^{\ast }\right)$ on the line segment joining A and B such that . $f\left(x_1,y_1\right)-f\left(x_0,y_0\right)=f_x\left(x^{\ast },y^{\ast }\right)\left(x_1-x_0\right)+f_y\left(x^{\ast },y^{\ast }\right)\left(y_1-y_0\right)$ This result is the two-dimensional version of the Mean-Value Theorem.