Derivative problem

The sum of two numbers is 12. Find these numbers if:
a) The sum of their third powers is minimal.
b) The product of one with the cube of the other is maximal.
c) Both are positive and the product of one with the other power of the other is maximal.

Correct result:

a1 =  6
b1 =  6
a2 =  3
b2 =  9
a3 =  4
b3 =  8


a+b=12 y=min(a3+b3) y=min(a3+(12a)3)  f=3a23(12a)2 f=0   3 x23 (12x)2=0  72x=432  x=6  a1=6
f2=a b3 f2=a (12a)3 f2(a)=d/da(a(12a)3)=4(a3)(12a)2  f2(a)=0  m1=3 m2=12 m3=12  f21=m1 (12m1)3=3 (123)3=2187 f22=m2 (12m2)3=12 (1212)3=0   a2=m1=3
f3=a b2 f3=a (12a)2 f3(a)=d/da(a(12a)2)=3(a216 a+48)  f3(a)=0  3(z216z+48)=0  3(z216 z+48)=0 3z248z+144=0  a=3;b=48;c=144 D=b24ac=48243144=576 D>0  z1,2=b±D2a=48±5766 z1,2=48±246 z1,2=8±4 z1=12 z2=4   Factored form of the equation:  3(z12)(z4)=0  f31=z1 (12z1)2=12 (1212)2=0 f32=z2 (12z2)2=4 (124)2=256  a3=z2=4

Our quadratic equation calculator calculates it.


We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

Showing 0 comments:

Tips to related online calculators

Next similar math problems:

  • Minimum of sum
    derive_1 Find a positive number that the sum of the number and its inverted value was minimal.
  • Three members GP
    exp_growth The sum of three numbers in GP (geometric progression) is 21 and the sum of their squares is 189. Find the numbers.
  • Find two
    eq222_7 Find two consecutive natural numbers whose product is 1 larger than their sum. Searched numbers expressed by a fraction whose numerator is the difference between these numbers and the denominator is their sum.
  • Secret treasure
    max_cylinder_pyramid Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.
  • Cube surfce2volume
    cube_shield Calculate the volume of the cube if its surface is 150 cm2.
  • Rectangle pool
    basen_5 Determine dimensions of open pool with a square bottom with a capacity 32 m3 to have painted/bricked walls with least amount of material.
  • One third power
    cube-root Which equation justifies why ten to the one-third power equals the cube root of ten?
  • 1 page
    books 1 page is torn from the book. The sum of the page numbers of all the remaining pages is 15,000. What numbers did the pages have on the page that was torn from the book?
  • Cube surface and volume
    cube_shield_6 Find the surface of the cube with a volume of 27 dm3.
  • Three numbers
    sigma Create from digits 1-9 three-digit numbers with their sum the smallest. What value is the sum of these numbers? (Use each digit only once)
  • Area to volume
    two_cubes_2 If the surface area of a cube is 486, find its volume.
  • Completing square
    eq2_5 Solve the quadratic equation: m2=4m+20 using completing the square method
  • The cube
    cube_shield_1 The cube has a surface area of 216 dm2. Calculate: a) the content of one wall, b) edge length, c) cube volume.
  • Minimum surface
    cuboid_20 Find the length, breadth, and height of the cuboid-shaped box with a minimum surface area, into which 50 cuboid-shaped blocks, each with length, breadth, and height equal to 4 cm, 3 cm, and 2 cm respectively, can be packed.
  • Cuboid and ratio
    cuboid_2 Find the dimensions of a cuboid having a volume of 810 cm3 if the lengths of its edges coming from the same vertex are in ratio 2: 3: 5
  • Cube V2S
    cube_shield The volume of the cube is 27 dm cubic. Calculate the surface of the cube.
  • Curve and line
    parabol The equation of a curve C is y=2x² -8x+9 and the equation of a line L is x+ y=3 (1) Find the x co-ordinates of the points of intersection of L and C. (2) Show that one of these points is also the stationary point of C?