$3-$b$)$ Fix a constant $Kin{R}^{+}.$ Prove that $x|(K-x{)}^{+}|$is convex.

$[$Hint: You need to consider three cases.$]$

$3-$c$)$ Consider an asset $S$ with derivatives of European type. Suppose, current prices of three call options are as follows

$C_{100}^E=$$$80,C_{145}^E=$$$36,C_{160}^E=$$$20$

Find an arbitrage. That is$,$ create a portfolio that results in an arbitrage at the maturity date $T.$ Interest rate is $r>0$ compounded continuously.