Application Of Derivatives

f(%5Ctheta)%3D%5Csqrt%7Ba%5E%7B2%7D%20%5Ccos%20%5E%7B2%7D%20%5Ctheta%2B%20b%5E%7B2%7D%20%5Csin%20%5E%7B2%7D%20%5Ctheta%7D%2B%5Csqrt%7Ba%5E%7B2%7D%20%5Csin%20%5E%7B2%7D%20%5Ctheta%20%2Bb%5E%7B2%7D%20%5Ccos%20%5E%7B2%7D%20%5Ctheta%7D

%5Ctherefore%5Bf(%5Ctheta)%5D%5E%7B2%7D%3Da%5E%7B2%7D%20%5Ccos%20%5E%7B2%7D%20%5Ctheta%20%2Bb%5E%7B2%7D%20%5Csin%20%5E%7B2%7D%20%5Ctheta%20%2Ba%5E%7B2%7D%20%5Csin%20%5E%7B2%7D%20%5Ctheta%20%2Bb%5E%7B2%7D%20%5Ccos%20%5E%7B2%7D%20%5Ctheta

%2B2%20%5Csqrt%7B%5Cleft(a%5E%7B2%7D%20%5Ccos%20%5E%7B2%7D%20%5Ctheta%20%2Bb%5E%7B2%7D%20%5Csin%20%5E%7B2%7D%20%5Ctheta%5Cright)%5Cleft(a%5E%7B2%7D%20%5Csin%20%5E%7B2%7D%20%5Ctheta%20%2Bb%5E%7B2%7D%20%5Ccos%20%5E%7B2%7D%20%5Ctheta%5Cright)%7D

%5Cbecause%5Bf(%5Ctheta)%5D%5E%7B2%7D

will be maximum, if

%5Csin%20%5E%7B2%7D%20%5Ctheta%3D%5Ccos%20%5E%7B2%7D%20%5Ctheta%3D%5Cfrac%7B1%7D%7B2%7D

and will be minimum, if either

%5Csin%20%5E%7B2%7D%20%5Ctheta%3D0

%5Ccos%20%5E%7B2%7D%20%5Ctheta%3D0

%5Ctherefore%20M%3D%5Cleft(a%5E%7B2%7D%2Bb%5E%7B2%7D%5Cright)%2B2%20%5Csqrt%7B%5Cleft(%5Cfrac%7Ba%5E%7B2%7D%7D%7B2%7D%2B%5Cfrac%7Bb%5E%7B2%7D%7D%7B2%7D%5Cright)%5E%7B2%7D%7D

%3D2%5Cleft(a%5E%7B2%7D%2Bb%5E%7B2%7D%5Cright)

and

m%3D%5Cleft(a%5E%7B2%7D%2Bb%5E%7B2%7D%5Cright)%2B2%20%5Csqrt%7Ba%5E%7B2%7D%20b%5E%7B2%7D%7D

%3Da%5E%7B2%7D%2Bb%5E%7B2%7D%2B2%20a%20b

%5Ctherefore%20M-m%3D2%5Cleft(a%5E%7B2%7D%2Bb%5E%7B2%7D%5Cright)-%5Cleft%5Ba%5E%7B2%7D%2Bb%5E%7B2%7D%2B2%20a%20b%5Cright%5D

%3Da%5E%7B2%7D%2Bb%5E%7B2%7D-2%20a%20b%3D(a-b)%5E%7B2%7D

Given,

f(x)%20%3Dx%5E%7B2%7D%2B2%20x%2B2

%3Dx%5E%7B2%7D%2B2%20x%2B1%2B1%3D(x%2B1)%5E%7B2%7D%2B1

Here,

f(x)%20%5Cin%5B1%2C%20%5Cinfty)

and

g(x)%3D-x%5E%7B2%7D%2B2%20x-1

%3D-%5Cleft(x%5E%7B2%7D-2%20x%2B1%5Cright)%3D-(x-1)%5E%7B2%7D

Here,

g(x)%20%5Cin(-%5Cinfty%2C%200%5D

Now,

%5Cfrac%7Bf%7D%7Bg%7D(x)%3D%5Cfrac%7Bx%5E%7B2%7D%2B2%20x%2B2%7D%7B-x%5E%7B2%7D%2B2%20x-1%7D%3Dy

%5CRightarrow%20x%5E%7B2%7D%2B2%20x%2B2%3D-y%20x%5E%7B2%7D%2B2%20x%20y-y

%5CRightarrow%20x%5E%7B2%7D%2By%20x%5E%7B2%7D%2B2%20x-2%20x%20y%2B2%2By%3D0

%5CRightarrow%20x%5E%7B2%7D(1%2By)%2B(2-2%20y)%20x%2B2%2By%3D0

%5Cbecause%20D%20%5Cgeq%200

%5Ctherefore%20(2-2%20y)%5E%7B2%7D-4(2%2By)(1%2By)%20%5Cgeq%200

4%2B4%20y%5E%7B2%7D-8%20y-4(2%2By)(1%2By)%20%3E%200

%5CRightarrow%20y%20%5Cleq-%5Cfrac%7B1%7D%7B5%7D

So,

%5Cfrac%7Bf%7D%7Bg%7D(x)%20%5Cin%5Cleft(-%5Cinfty%2C-%5Cfrac%7B1%7D%7B5%7D%5Cright%5D

So,

a%3D1%2C%20b%3D0

and

c%3D-%5Cfrac%7B1%7D%7B5%7D

Hence,

a%2B2%20b%2B5%20c%2B4

%3D1%2B0%2B5%5Cleft(-%5Cfrac%7B1%7D%7B5%7D%5Cright)%2B4

%3D1-1%2B4%3D4

Chapter Questions
2 MCQs