Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/267



Given $$f(x, y) \equiv Ax^2 + Bxy + Cy^2$$, expand $$f(x + h, \ y + k)$$ in powers of $$h$$ and $$k$$.

The third and higher partial derivatives are all zero. Substituting in (67), $$\begin{align} f\left( x + h,y + k \right) \equiv & Ax^2 + Bxy + Cy^2 + \left( 2 Ax + By \right) h + \left( Bx + 2 Cy \right) k \\ & + Ah^2- + Bhk + Ck^2. \end{align}$$ Ans. Given $$f(x, y, z) \equiv Ax^2 + By^2 + Cz^2$$, expand $$f(x + l,\ y + m, z + n)$$ in powers of $$l$$, $$m$$, $$n$$.

The third and higher partial derivatives are all zero. Substituting in (68), $$\begin{align} f\left(x + l, y + m, z + n \right) \equiv & Ax^2 + By^2 + Cz^2 + 2 Axl + 2 Bym + 2 Czn \\ & + Al^2 + Bm^2 + Cn^2. \\ \end{align}$$ Ans.

Given $$f(x, y) \equiv \sqrt{x} \tan y$$, expand $$f(x + h, y + k)$$ in powers of $$h$$ and $$k$$. Given $$f(x,\ y,\ z) \equiv Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fzx$$, expand $$f(x + h,\ y + fc,\ z + l)$$ in powers of $$h$$, $$k$$, $$l$$.

The function $$f(x, y)$$ is said to be a maximum at $$x = a,\, y = b$$ when $$f(a, b)$$ is greater than $$f(x, y)$$ for all values of $$x$$ and $$y$$ in the neighborhood of $$a$$ and $$b$$. Similarly, $$f(a, b)$$ is said to be a minimum at $$x = a,\, y = b$$ when $$f(a, b)$$ is less than $$f(x, y)$$ for all values of $$x$$ and $$y$$ in the neighborhood of $$a$$ and $$b$$. These definitions may be stated in analytical form as follows: If, for all values of $$h$$ and $$k$$ numerically less than some small positive quantity,


 * {| style="width:100%"


 * style="width:3%"|
 * align="left" |$$f(a + h, b + k) - f(a, b )= $$ a negative number, then $$f(a,b)$$ is a maximum value of $$f(x,y)$$.
 * }

If


 * {| style="width:100%"


 * style="width:3%"|
 * align="left" |$$f(a + h, b + k) - f(a, b )= $$ a positive number, then $$f(a,b)$$ is a minimum value of $$f(x,y)$$.
 * }

These statements may be interpreted geometrically as follows: a point $$P$$ on the surface