Prove that $e^{tx} \le xe^t + 1-x$

Question

Prove that $$e^{tx} \le xe^t + 1-x$$ for $t \ge 1$ and $0 \le x \le 1$

I think I need to use the fact that e is convex? But I can't quite see it.

Any help appreciated Thanks.

Answer 1

$$ e^{tx} = e^{tx+(1-x)0} \leq xe^t + (1-x)e^0, $$ by convexity.