Problems

1. For each of the following functions:
   1. Calculate the derivative,
   2. Graph the function,
   3. Use your calculations in (i) to identify the range of values of `t` for which the graph is increasing, the range of values of `t` for which the graph is decreasing, and all local maxima and local minima.
      

   `ftext[(]t text[)]=e^(4t)`	`utext[(]t text[)]=t+1`
   `gtext[(]t text[)]=e^(-3t)`	`vtext[(]t text[)]=t^3+t^2+t+1`
   `htext[(]t text[)]=3^t-t^3`	`wtext[(]t text[)]=t^2+t+1/t`

2. Show that `3x^2-3x+1` is never negative. (Graphing the function is not enough. You have to justify your answer.)
3. Find the number `x` between 0 and 4 where `ftext[(] x text[)]=x^3-3x^2+9x+5` is the smallest. Explain your answer.
4. We have seen that the derivative of the reciprocal function, `1 text[/] t`, is the function `-1 text[/] t^2`, which is always negative. That would seem to suggest that the reciprocal function is always decreasing. But the values of `1 text[/] t` are negative when `t` is negative and positive (bigger than negative) when `t` is positive. Explain this apparent contradiction — carefully.
5. 1. Find the minimum value of the function
             `ftext[(] x text[)]=2|x-1|-|x-2|+1/5 x^2`.
   2. Explain why the minimum does not occur at a value of `x` for which `f'text[(] x text[)]=0`.
   3. Graph the function.