4.10 More on partial derivatives  (Page 3/3)

Let $f$ be defined on $R^2$ by $f(0,0)=0$ and, for $(x,y)\ne (0,0),$ $f(x,y)=x^{3}y/(x^2+y^2).$

1. Prove that both partial derivatives $f_x$ and $f_y$ exist at each point in the plane.
2. Show that $f_{yx}(0,0)=1$ and $f_{xy}(0,0)=0.$
3. Show that $f_{xy}$ exists at each point in the plane, but that it is not continuous at $(0,0).$

Let $f:S\to R$ be a real-valued function of two real variables, and let $c=(a,b)\in S^0$ be a point at which $f$ attains a local maximum or a local minimum. Show that ifeither of the partial derivatives $tialf/tialx$ or $tialf/tialy$ exists at $c,$ then it must be equal to 0.

HINT: Just consider real-valued functions of a real variable like $x\to f(x,b)$ or $y\to f(a,y),$ and use [link] .

1. Finish the proof to the preceding theorem by showing that
   $$\underset{h\to 0}{lim}\frac{{\theta }_f(k_1\left(h\right),k_2\left(h\right))}{h}=0.$$
   HINT: Review the corresponding part of the proof to [link] .
2. Suppose $f:S\to R$ is as in the preceding theorem and that $\phi$ is a real-valued function of a real variable.Suppose $f$ is differentiable, as a function of two real variables, at the point $(a,b),$ and that $\phi$ is differentiable at the point $c=f(a,b).$ Let $g=\phi \circ f.$ Find a formula for the partial derivatives of the real-valued function $g$ of two real variables.
3. (A generalized Mean Value Theorem) Suppose $u$ is a real-valued function of two real variables, both of whose partial derivatives exist at each point in a disk $B_r(a,b).$ Show that, for any two points $(x,y)$ and $(x^{\text{'}},y^{\text{'}})$ in $B_r(a,b),$ there exists a point $(\widehat{x},\widehat{y})$ on the line segment joining $(x,y)$ to $(x^{\text{'}},y^{\text{'}})$ such that
   $$u(x,y)-u(x^{\text{'}},y^{\text{'}})=\frac{tialu}{tialx}(\widehat{x},\widehat{y})(x-x^{\text{'}})+\frac{tialu}{tialy}(\widehat{x},\widehat{y})(y-y^{\text{'}}).$$
   HINT: Let $\phi :[0,1]\to R^2$ be defined by $\phi \left(t\right)=(1-t)(x^{\text{'}},y^{\text{'}})+t(x,y).$ Now use the preceding theorem.
4. Verify that the assignment $f\to tialf/tialx$ is linear; i.e., that
   $$\frac{tial(f+g)}{tialx}=\frac{tialf}{tialx}+\frac{tialg}{tialx}.$$
   Check that the same is true for partial derivatives with respect to $y.$