Optimal bid for a company worth double to you

A company's current value $V$ is uniformly distributed on $[0,1]$; you do not know $V$, but the owner does. You make a single take-it-or-leave-it offer $b$. The owner accepts only if $b\ge V$, and once you own the firm its value becomes $2V$. Which bid $b$ maximises your expected profit?

Every bid earns zero in expectation, so you gain nothing by bidding (bid $0$). An accepted bid $b$ tells you $V\sim U[0,b]$, so the firm is worth $2\cdot\tfrac b2=b$ to you on average, exactly what you paid: expected profit is $\int_0^b(2v-b)\,dv=0$. With any value uplift below $100\%$ every positive bid loses money (the winner's curse).