Q. 53. - Calculus

Question

For the partial derivatives given in Exercises 51–54, find the

most general form for a function of two variables, , with

the given partial derivative

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

Step-by-Step Solution

Answer

The required most general form of $f(x,y)$ so that $\frac{d^{2}f}{d{x}^2}=0$  is $f(x,y)=x{h}_1\left(y\right)+h_2\left(y\right)$

Step 1: Given information

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

Step 2: The objective is to find the most general form of a function f ( x ,   y )  

The most general form of a function $f(x, y)$so that $\frac{{\partial }^{2}f}{\partial x^2}=0$

Suppose, $f(x,y)=x{h}_1\left(y\right)+h_2\left(y\right)$

Then,

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

Hence, the most general form of $f(x, y)$so that $\frac{{\partial }^{2}f}{\partial x^2}=0$is $f(x,y)=x{h}_1\left(y\right)+h_2\left(y\right)$