Suppose that the function $f ( x , y )$ and the linear function $L ( x , y ) = 3 - 3 x + 4 y$ satisfy $| f ( x , y ) - L ( x , y ) | \leq 5 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 }$ for points $( x , y )$ close to $( 0,0 )$ .Is f differentiable at (0,0)?
A)Yes
B)No

Question

Quizplus · Accepted Answer

The given inequality $| f ( x , y ) - L ( x , y ) | \leq 5 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 }$ implies that the difference between $f(x, y)$ and the linear function $L(x, y)$ approaches 0 faster than the distance from $(x, y)$ to $(0, 0)$ approaches 0, as $(x, y)$ approaches $(0, 0)$. This behavior is consistent with $f(x, y)$ being differentiable at $(0, 0)$, with $L(x, y)$ serving as the linear approximation (or derivative) of $f$ at that point.

Suppose That the Function f(x,y)f ( x , y )f(x,y) And the Linear Function

Suppose That the Function $f ( x , y )$ And the Linear Function