Which of the following is/are true: Which of the following is/are true:


*

*$\log\dfrac{x+y}{2}\le\dfrac{\log x+\log y}{2}~\forall~x,y>0;$

*${e}^{\frac{x+y}{2}}\leq\dfrac{e^x+e^y}{2}~\forall~x,y>0.$
 A: Hint: Use Jensen's inequality.
A: The second one is true, since $\exp(x)$ is concave upward in any interval $I$ and for all $x,y\in I$ such that $x<y$. Note that $$\exp(a)<\frac{y-a}{y-x}\exp(x)+\frac{a-x}{y-x}\exp(y)$$ where $x<a<y$.
A: The question can be reformulated as "Which of the following is/are true:

(1) $\log x$ is midpoint convex.
(2) $e^x$ is midpoint convex."

For real-valued continuous functions on connected subsets of $\Bbb R$, midpoint convexity is equivalent to convexity. When such a function is differentiable (as in these cases), we can show convexity holds by showing that the second derivative is nonnegative on the whole domain, and we can show that convexity fails by finding a point in the domain where the second derivative is negative. (Of course, you may not have differentiation in your toolbox, yet.)
It's worth noting that the two functions are inverses of each other, so one of them is convex if and only if the other is concave. The only functions that are both convex and concave are linear functions. Hence, at most one of these is true.
A: 1.$x=1, y=2$
2.Applying AM-GM inequality.
