HW 33, Additional Problems:
 
1.
Let f(x) = ex(x3 - 3x2 + 5x -5).
(a)
Find all the critical points of f. Classify each one as either a local maximum, local minimum, or point of inflection.
(b)
Does f have an absolute maximimun value? If so, what is it and where is it attained?
(c)
Does f have an absolute minimimun value? If so, what is it and where is it attained?
2.
Let $f(x)=\sqrt{(-x^2 + 4)^2}$.
Please note that $f(x)\ne -x^2 + 4$. For instance, if x = 10 f(x) is positive, but -x2 + 4 is negative.
(a)
Differentiate f using the Chain Rule. Do not try to simplify first.
(b)
What are the critical points of f? Classify them.
3.
You are in a dune buggy 40 km north of the nearest point (we'll call it P) on a straight paved road. 50 km down the road from P is the nearest gas station. The dune buggy is low on gas and it's late in the afternoon, so you want to get to the gas station in the minimum amount of time.
(a)
Suppose your dune buggy can travel 45 k/min the desert and 75 k/min a paved road. If you take the route that minimizes time, how long will it take you to get to the gas station?
(b)
Suppose your dune buggy can travel 45 k/min the desert and 55 k/min a paved road. If you take the route that minimizes time, how long will it take you to get to the gas station?

 

 

Extra credit:
 
(a)
Let $f(x)=\sqrt{(x^2-1)^2} + x$.
Please note that $f(x)\ne x^2 - 1 + x$. For instance, if x = 0 f(x) is 1, not -1.
i.
Differentiate f using the Chain Rule. Do not try to simplify first.
ii.
What are the critical points of f? Classify them.



updated 12/17/1999

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