Question Bank (unit-6)

Question-1: If x = r sin

                      sin  also find 

Sol.  We know that,

                                       =

                                       =

                                       =      ( on solving the determinant)

                                      = 

Now using first propert of Jacobians, we get

Question-2: If u = x + y + z , uv = y + z , uvw = z ,  find

Sol.  Here we have,

              x = u – uv = u(1-v)

             y = uv – uvw = uv( 1- w)

And      z = uvw

So that,

                                                         =

Apply

                             =                

Now we get,

                          =  u²v(1-w) + u²vw

                          = u²v

Question-3: If u = xyz , v = x² + y² + z²  and w = x + y + z, then find   J = 

Sol. Here u ,v and w are explicitly given , so that first we calculate

                              J’ =

                             J’ =   =

  = yz(2y-2z) – zx(2x – 2z) + xy (2x – 2y) = 2[yz(y-z)-zx(x-z)+xy(x-y)]

  = 2[x²y - x²z - xy² + xz² + y²z - yz²]

  = 2[x²(y-z)  - x(y² - z²) + yz (y – z)]

  = 2(y – z)(z – x)(y – x)

  = -2(x – y)(y – z)(z – x)

By the property,

                            JJ’ =  1

                         J =

Question-4: If  w = x² + y – z + sin t and x + y = t then find

Sol.  With x, y and z independent , we have

Therefore , we get

                                              = 2x + cos ( x+ y)

Question-5: Find out the maxima and minima of the function

Sol   Given     …(i)

         Partially differentiating (i) with respect to x we get

                   ….(ii)

         Partially differentiating (i) with respect to y we get

                    ….(iii)

          Now, form the equations

                    Using (ii) and (iii) we get

                                  using above two equations

                     Squaring both side we get

                      Or 

                       This show that

                         Also we get

                 Thus we get the pair of value as

                  Now, we calculate

      Putting above values in

      At point (0,0)  we get

      So, the point (0,0) is a saddle point.

      At point we get

        So the point is the minimum point where

          In case 

         So the point is the maximum point where

Question-6: Find the maximum and minimum point of the function

  Sol.

Partially differentiating given equation with respect to and x and y then equate them to zero                 

       On solving above we get 

         Also

            Thus we get the pair of values (0,0), (,0) and (0,

        Now, we calculate

        At the point (0,0)

       So function has saddle point at (0,0).

       At the point (

         So the function has maxima at this point (.

        At the point (0,

       So the function has minima at this point (0,.

        At the point (

        So the function has an saddle point at (

Question-7: Find the maximum and minimum value of

Sol.

Let

Partially differentiating given function with respect to x and y and equate it to zero

          ..(i)

           ..(ii)

      On solving (i) and (ii) we get

Thus pair of values are

Now, we calculate

At the point (0,0)

  So further investigation is required

On the x  axis y  = 0  , f(x,0)=0

On the line y=x,

At the point

So that the given function has maximum value at

Therefore maximum value of given function

At the point

So that the given function has minimum value at

Therefore minimum value of the given function

Question-8: Divide 24 into three parts such that the continued product of the first, square of second and cube of third may be maximum.

Sol.

Let first number be x, second be y and third be z.

According to the question

Let the  given function  be  f

And the relation

By Lagrange’s Method

   ….(i)

Partially differentiating (i) with respect to x,y and z and  equate  them  to zero

   ….(ii)

   ….(iii)

   ….(iv)

From (ii),(iii)  and (iv) we get

On solving

Putting it in given relation we get

Or 

Or

Thus the first number is 4 second is 8 and third is 12

Question-9: If ,Find the value of x and y for  which is maximum.

Sol.

Given function is

And relation is

By Lagrange’s Method

[]    ..(i)

Partially differentiating (i) with respect to  x, y and z and equate them tozero

Or       …(ii)

Or       …(iii)

Or       …(iv)

On solving (ii),(iii) and (iv) we  get

Using the given relation  we get

So that

Thus the point for the maximum value of the given function is

Question-10: Find  the points on the surface nearest to the origin.

Sol.

Let be any point on the surface, then its distance from the origin is

Thus the given equation will be

And relation is

By Lagrange’s Method

     ….(i)

Partially differentiatig (i) with respect to  x, y and z and equate them to zero

Or        …(ii)

Or        …(iii)

Or

Or

On solving equation  (ii) by (iii) we get

And     

On subtracting we  get

Putting in above

Or 

Thus

Using the given relation we  get

= 0.0 +1=1

Or 

Thus point on the surface nearest to the origin is