maximum and minimum problems

debanik2 · P versus NP Offline Joined: 24 Apr 2009 Posts: 40

does anyone have interesting problems on maxima and minima?

**Posted:** Sat Nov 07, 2009 5:28 am

akech

Every calculus textbook has a section on Applications of Differentiation!

1. Find the absolute maximum value of the function: $g(t) = \frac{1}{1 + |t|} + \frac{1}{1 + |t-2|}$

2. Find the local maximum and minimum values for the function: $f(x) = e^{-\frac{1}{x+1}}$

**Posted:** Sat Nov 07, 2009 1:33 pm

Kent Merryfield

Here's one I just assigned to my Putnam problems class. It's Putnam 1996 A1:

Find the least number $A$ such that for any two squares of combined
area 1, a rectangle of area $A$ exists such that the two squares can
be packed in the rectangle (without interior overlap). You may assume
that the sides of the squares are parallel to the sides of the
rectangle.

**Posted:** Sat Nov 07, 2009 5:21 pm

AndrewTom

The Putman question is very nice.

For 2, it seems that the function approaches $\infty$ as $x$ approaches $- 1$ from the left and that it approaches $0$ as $x$ approaches $- 1$ from the right.

For 1, sketching the graph, considering $x < 0, x = 0, 0 < x < 2, x = 2$ and $x > 2$ , we see that there is a global maximum of $\frac {4}{3}$ when $x = 0$ .

3. Prove that a solid right circular cone of given total surface area has its greatest volume when the slant height is three times the radius of the base.

**Posted:** Sun Nov 08, 2009 7:19 am