Answer
$\dfrac{\sqrt{145}}{2},\lt 4,\dfrac{9}{2} \gt$
Work Step by Step
Our aim is to determine the maximum rate of change of $f(x,y)$. In order to find this, we have : $D_uf=|\nabla f(x,y)|$
Given: $f(x,y)=x^2y+\sqrt y$
$\nabla f(x,y)=\lt 2xy, x^2+\dfrac{1}{2\sqrt y} \gt$
From the given data, we have $f(x,y)=f(2,1)$
$\nabla f(2,1)=\lt (2)(2)(1),2^2+\dfrac{1}{2\sqrt 1} \gt=\lt 4,\dfrac{9}{2} \gt$
$|\nabla f(2,1)|=\sqrt{4^2+(\dfrac{9}{2})^2}=\dfrac{\sqrt{145}}{2}$
Therefore, the maximum rate of change of $f(x,y)$ and the direction is: $\dfrac{\sqrt{145}}{2},\lt 4,\dfrac{9}{2} \gt$
