Two numbers sum to $100$. What is the smallest possible value of the sum of one number and the square of the other number?

Business and Economics

60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16

56. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the accompanying figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax^3+bx^2+cx+d, where y(-L)= H and y(0)=0.

a. What is dy/dx at x = 0?

b. What is dy/dx at x = -L?

c. Use the values for dy/dx at x = 0 and x =- L together with y(0) = 0 and y(-L) = H to show that y(x)=H[2(x/L)^3+3(x/L)^2]

54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)

Suppose the product of two positive real numbers is $16$. Which pair of numbers has the smallest possible sum?

For the function $y=3x^4-2x$ defined on the closed interval $[0,1]$, at what value of $x$ does the minimum value of $y$ occur?

Given the region bounded by the x-axis, the y-axis, and the line $y=4-x$ in the first quadrant, determine the $x$ and $y$ coordinates of the centroid of the shaded area.

Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the constraint $x+y=4$.