Calculus/max/min

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Expert: - 10/26/2008


Question
A triangle is formed by a line through (4,1) and the two coordinate axes.
a) Is there a minimal area for such a triangle? If so, what is it? If not, why not?
b) Is there a maximal area for such a triangle? If so, what is it? If not, why not?

Answer
The line passing through (4,1) has the equation
y-1=m(x-4) and sope m.

To find the y intercept, set x to 0 and solve for y.
You should get y = 1 - 4m.

To find the x intercept, set y to 0 and solve for x.
You should get x = 4 - 1/m.

Multiply the value gotten for x by the value gotten for y,
then divide the entire answer by 2.  
This is the equation for the area of the triangle.

Take the derivative with respect to m.

a) You should find a value of m that gives a minimum.

b) The closer y gets to 1 (from above), the bigger x gets.  There is no maximum.  

Calculus

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