3.3 Multivariable Differential and Integral Calculus 171f (x, y)=x^4 + 6 x^2 y^2 +y^4 −9
4
x−7
4
y.503.Find the equation of the smallest sphere that is tangent to both of the lines (i)x=
t+1,y= 2 t+4,z=− 3 t+5, and (ii)x= 4 t−12,y=−t+8,z=t+17.
504.Determine the maximum and the minimum of cosA+cosB+cosCwhenA, B,
andCare the angles of a triangle.
505.Prove that forα, β, γ∈[ 0 ,π 2 ),
tanα+tanβ+tanγ≤2
√
3
secαsecβsecγ.506.Prove that any real numbersa, b, c, dsatisfy the inequality
3 (a^2 −ab+b^2 )(c^2 −cd+d^2 )≥ 2 (a^2 c^2 −abcd+b^2 d^2 ).When does equality hold?507.Givennpoints in the plane, suppose there is a unique line that minimizes the sum
of the distances from the points to the line. Prove that the line passes through two
of the points.
To find the maximum of a function subject to a constraint we employ the following
theorem.
The Lagrange multipliers theorem.If a functionf (x, y, z)subject to the constraint
g(x, y, z)=Chas a maximum or a minimum, then this maximum or minimum occurs at
a point(x,y,z)of the setg(x, y, z)=Cfor which the gradients offandgare parallel.
So in order to find the maximum offwe have to solve the system of equations
∇f =λ∇gandg(x, y, z)=C. The numberλis called the Lagrange multiplier; to
understand its significance, imagine thatfis the profit andgis the constraint on resources.
Thenλis the rate of change of the profit as the constraint is relaxed (economists call this
the shadow price).
As an application of the method of Lagrange multipliers, we will prove the law of
reflection.
Example.For a light ray reflected off a mirror, the angle of incidence equals the angle of
reflection.
Solution.Our argument relies on the fundamental principle of optics, which states that
light travels always on the fastest path. This is known in physics as Fermat’s principle
of least time. We consider a light ray that travels from pointAto pointBreflecting