$(5)$ Assume that $g$ is a convex function on $R^n,$ that $f$ is a linear function of a single variable,

and in addition that $f$ is a nondecreasing function $($which means that $f(r)f(s)$ whenever

$rs.$

$($a$)$ Show that $F$:$=f$@$g$ is convex by directly verifying the convexity inequality

$F(x+(1-)y)F(x)+(1-)F(y).$

Explain where each hypothesis $($convexity of $g,$ linearity of $f,$ and the fact that $f$ is

nondecreasing$)$ is used in your reasoning. $($The notation $F=f$@$g$ means that $F(x)=f(g(x)).)$Discussion: Expressing $gra{d}^{2}F$ in terms of $f$ and $g$ is basically an exercise in using the chain rule for functions of

several variables. If you find it at all difficult, then review the chain rule until you have completely mastered

it$!$ When showing that $gra{d}^{2}F$ is positive semidefinite, please explain again, as you did in part $($a$),$ where each

hypothesis is used in your reasoning.$(5)$ Assume that $g$ is a convex function on $R^n,$ that $f$ is a linear function of a single variable,

and in addition that $f$ is a nondecreasing function $($which means that $f(r)f(s)$ whenever

$rs.$

$($a$)$ Show that $F$:$=f$@$g$ is convex by directly verifying the convexity inequality

$F(x+(1-)y)F(x)+(1-)F(y).$

Explain where each hypothesis $($convexity of $g,$ linearity of $f,$ and the fact that $f$ is

nondecreasing$)$ is used in your reasoning. $($The notation $F=f$@$g$ means that $F(x)=f(g(x)).)(6)$ Let $S=\{(x,y)in{R}^2|3y6\}sub{R}^2$ be the strip between two lines in the plane. Let

$d$:$SR$ be the distance from a point $(a,b)inS$ to the boundary:

$d(a,b)=mi{n}_{(x,y)indelS}||(a,b)-(x,y)||.$

Show that $d$ is a concave function on $S.$ Hint: Find a formula for $d.$