12 In time t, a tree can grow to a size F(t), where F (t)

0 and F(t) 0. Assume that for large t, F (t) is near 0. If the tree is cut at time t, then a revenue F(t) is received.

Assume that revenues are discounted continuously at a rate r, so $1 received at time t is equivalent to $ert received at time 0. The goal is to cut the tree at the time t* that maximizes discounted revenue. Show that the tree should be cut at the time t* satisfying the equation

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F'(*) r F(t*) In the answer, explain why (if>r) this equation has a unique solution. Also show that the answer is a maximum, not a minimum. [Hint: Why is it sufficient to choose t* to maximize In(e "F(t)?]