7. Find the directions in which the function increases and decreases most rapidl

Question

7. Find the directions in which the function increases and decreases most rapidly at Po- Then find the derivatives of the function in those directions. f(x,yz)-3 In (xy) + 4 In (yz) + In (xz), Po (1,1,1 ) In which direction does the function increase most rapidly? (Type exact answers, using radicals as needed.) In which direction does the function decrease most rapidly? i+ -u= (Type exact answers, using radicals as needed.) What is the value of the derivative in the direction of u? (Type an exact answer, using radicals as needed.) What is the value of the derivative in the direction of -u? (Type an exact answer, using radicals as needed.)

Explanation / Answer

7)

f(x,y,z)=3ln(xy) +4ln(yz) +ln(xz)

f(x,y,z)=3(ln(x)+ln(y)) +4(ln(y)+ln(z)) +(ln(x)+ln(z))

f(x,y,z)=3ln(x)+3ln(y) +4ln(y)+4ln(z) +ln(x)+ln(z)

f(x,y,z)=4ln(x)+7ln(y)+5ln(z)

f=(4/x)i +(7/y)j +(5/z)k

at P0(1,1,1)

f=(4/1)i +(7/1)j +(5/1)k

f=4i +7j +5k

|f|=(42+72+52)

|f|=90

|f|=310

u=f/|f|

u=(4i +7j +5k)/310

u=(4/310)i +(7/310)j +(5/310)k

-u=(-4/310)i +(-7/310)j +(-5/310)k

(Duf)Po =f.u

(Duf)Po =(4i +7j +5k).((4/310)i +(7/310)j +(5/310)k)

(Duf)Po =((4*4/310) +(7*7/310) +(5*5/310))

(Duf)Po =((16/310) +(49/310) +(25/310))

(Duf)Po =(90/310)

(Duf)Po =310

(D-uf)Po =f.-u

(D-uf)Po=(4i +7j +5k).((-4/310)i +(-7/310)j +(-5/310)k)

(D-uf)Po =((-4*4/310) +(-7*7/310) +(-5*5/310))

(D-uf)Po =((-16/310) +(-49/310) +(-25/310))

(D-uf)Po =(-90/310)

(D-uf)Po =-310

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