Since , for all , we have .1. If , find .

Then using the chain-rule we find .

Write it as . Then (again) by the chain-rule,2. Let , where and k, b are non-zero constants.

Note that it's equavalent to - hence and . Then3. For , find the minimum value of .

we have . Recall that a minimum of is a point such that

and . Take the first zero of , that is , and we have ,

so it is a minimum. Thus the minimum value of is .

By the chain-rule (again) the derivative of [tex]e^{\frac{x}{3}}[/tex] is (fill the details). Since for any4. Find if

positive real , we have and the derivative of that is

(obviously) . Thus . (It's late over here so I apologise if there happen to be any

'mistypes' or mistakes).