Derivatives and graphing; please show all work. See attached.
#25 Using the answer to previous problem as a guide, write a short paragraph (using complete sentences) which describe the relationships between the following features of a function f:
a) the local maxima and minima of f
b) the points at which the graph of f changes concavity
c) the sign changes of f'
d) The local maxima and minima of f'
Pg 171 #26
The function f has a derivative everywhere and has just one critical point, at x=3. In parts (a) thru (d) are given additional conditions. In each case decide whether x=3 is a local maximum, local minimum or neither. Explain reasoning. Sketch possible graphs for all four cases.
a) f ' (1) = 3 and f ' (5) = -1
b) f(x) ? infinity as x ? infinity and as x ? -infinity
c) f(1) = 1, f(2)=2, f(4)=4, f(5)=5
d) f ' (2) = -1, f(3) = 1, f(x) ? 3 as x ? infinity
Pg 182 #29
An apple tree produces, on average, 400kg of fruit each season. However, if more than 200 trees are planted per square km, crowding reduces the yield by 1kg for each tree over 100.
a) Express the total yield from one sq kilometer as a function of the number of trees on it. Graph this function.
b) How many trees should a farmer plant on each square kilometer to maximize yield?
This shows how to determine maxima and minima for equations, including in a word problem.