Let $$ be a strictly concave differentiable function. Show that if $>>>$ and: $()+(1)()=()+(1)(),$ then $+(1)>+(1).$ Hint: Start by rearranging the above equality $($put the x$$s on one side and the y$$s on the other$)$ and rewriting it using the Mean Value Theorem. $($You should have learned what the Mean Value Theorem is in MATH $157.$ If you forgot what it is$,$ you can find the statement of the Mean Value Theorem on the internet in a few seconds.$)$ Then rearrange the inequality in the same way and compare it with the rewritten equality. Note: The above implies that when a risk$-$averse agent bears more risk $($note that outcome pair $(,)$ is riskier than outcome pair $(,)),$ then to make sure her expected utility stays the same, her expected income must increase. This observation will be useful when we study moral hazard