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Chapter 12: Q. 74 (page 946)

Letp(x)beapolynomialofdegreen,q(y)beanyfunctionofy,andf(x,y)=p(x)q(y).Provethat鈭�n+1f鈭�xn+1=0

Short Answer

Expert verified

鈭�n+1f鈭�xn+1=0

Step by step solution

01

Step 1. Given information

f(x,y)=p(x)q(y),where,p(x)isapolynomialofdegreen,q(y)isanyfunctionofy.

02

Step 2. Proofs of given partial derivatives 

LHS=鈭�n+1f鈭�xn+1=鈭�鈭�x鈭�np(x)q(y)鈭�xnUsingdefinitionofpartialderivatives,functionofyistreatedasaconstantterm=q(y)鈭�鈭�x鈭�np(x)鈭�xnUsingpowerrule,afterdifferentiatingpolynomialofdegreenforntimeswegetaconstantterm,whosedifferentiationgiveszeroasthefinalresultaftern+1timesofdifferentiationintotal.=q(y)鈭�K鈭�x=q(y)(0)=0=RHS

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Most popular questions from this chapter

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

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Let T be a triangle with side lengths a, b, and c. The semi-perimeter of T is defined to be s=12a+b+cHeron鈥檚 formula for the area A of a triangle is

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Use Heron鈥檚 formula and the method of Lagrange multipliers to prove that, for a triangle with perimeter P, the equilateral triangle maximizes the area.

In Exercises 21-28, find the directional derivative of the given function at the specified point Pand in the direction of the given unit vectoru .

f(x,y)=yxat P=(4,9),u=-1717,-41717

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Why does the method of Lagrange multipliers fail with this function?
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