91Ó°ÊÓ

Find the equation of the tangent of the curve $$ y=x^{3}-x $$ at the origin; at the point where it crosses the positive axis of $x .

Test the following curves for maxima, minima, and points of inflection, and determine the slope of the curve in each point of inflection. $$ y=4 x^{3}-15 x^{2}+12 x+1 $$

A farmer wishes to fence off a rectangular pasture along a straight river, one side of the pasture being formed by the river and requiring no fence. He has barbed wire enough to build a fence \(1000 \mathrm{ft}\). long. What is the area of the largest pasture of the above description which he can fence off?

Show that, of all rectangles having a given perimeter, the square has the largest area.

Divide the number 12 into two parts sueh that the sum of their squares may be as small as possible. (What is meant is such a division as this: one part might be 4 , and then the other would be 8. The sum of the squares would here be \(16+64=80 .\) )

Show that the area of the triangle formed by the coordinate axes and the tangent of the hyperbola at any point is constant. $$ x y=a^{2} $$

What is the shortest distance from the point \((10,0)\) to the parabola \(\quad y^{2}=4 x ?\)

Show that the portion of the tangent of the curve $$ x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} $$ at any point, intercepted between the coordinate axes, is constant.

Suggestion. Show that the derivative has no real roots and hence, being continuous, never changes sign. $$ y=1+2 x+x^{2}-x^{3} $$

Find the least value of the function $$ y=x^{2}+6 x+10 $$