Q. 58. - Calculus

Question

For the partial derivatives given in Exercises 55–58, find the

most general form for a function of three variables, $f(x,y,z)$,

with the given partial derivative.

$\frac{{\partial }^{2}f}{\partial y\partial x}=0$

Step-by-Step Solution

Answer

 The most general form of $f$ so that $\frac{{\partial }^{2}f}{\partial y\partial x}=0$is $f(x,y,z)=x{h}_1\left(z\right)+z{h}_2\left(y\right)$

Step 1: Given information

Given derivative is $\frac{{\partial }^{2}f}{\partial y\partial x}=0$

Step 2: The objective is to find the most general form of a function f ( x ,   y ,   z )   so that ∂ 2 f ∂ y ∂ x = 0  

Suppose, $f(x,y,z)=x{h}_1\left(z\right)+z{h}_2\left(y\right)$

Then,

$$\begin{gathered} \frac{df}{dx}=h_1\left(z\right)+0 \\ \frac{d^{2}f}{dydx}=0 \end{gathered}$$

Hence, the most general form of $f$ so that $\frac{{\partial }^{2}f}{\partial y\partial x}=0$is $f(x,y,z)=x{h}_1\left(z\right)+z{h}_2\left(y\right)$