I think ベストを尽くせ sounds a little too strong to encourage someone. In a friendly way, I would say がんばって。信じてるよ。 If you want to make it more polite, you want to say がんばってください。信じています。
@JPChina ベストを尽くせ sounds like an order and an oppressive way. Of course you can say like this, for example .... I might say this to my close friend right before a sports competition to cheer him/her up. But usually I would choose more mild way to say it ; がんばって. 信じてるよ is a friendly way that exactly means" I believe in you. " Of course you can say that.