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 answer:

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

Step-by-step explanation:

a+b=12 y=min(a3+b3) y=min(a3+(12−a)3)  f′=3a2−3(12−a)2 f′=0  3⋅ x2−3⋅ (12−x)2=0  72x=432  x=6  a1=6
b1=12−a1=12−6=6
f2=a⋅ b3 f2=a⋅ (12−a)3 f2′(a)=d/da(a(12−a)3)=−4(a−3)(12−a)2  f2′(a)=0  m1=3 m2=12 m3=12  f21=m1⋅ (12−m1)3=3⋅ (12−3)3=2187 f22=m2⋅ (12−m2)3=12⋅ (12−12)3=0  a2=m1=3
b2=12−a2=12−3=9
f3=a⋅ b2 f3=a⋅ (12−a)2 f3′(a)=d/da(a(12−a)2)=3(a2−16 a+48)  f3′(a)=0  3(z2−16z+48)=0  3(z2−16 z+48)=0 3z2−48z+144=0  a=3;b=−48;c=144 D=b2−4ac=482−4⋅3⋅144=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(z−12)(z−4)=0  f31=z1⋅ (12−z1)2=12⋅ (12−12)2=0 f32=z2⋅ (12−z2)2=4⋅ (12−4)2=256  a3=z2=4

Our quadratic equation calculator calculates it.

b3=12−a3=12−4=8

Try another example


Did you find an error or inaccuracy? Feel free to write us. Thank you!






avatar




Tips to related online calculators
Looking for help with calculating roots of a quadratic equation?

You need to know the following knowledge to solve this word math problem:

Related math problems and questions:

  • The sum
    eq222 The sum of the squares of two immediately following natural numbers is 1201. Find these numbers.
  • Minimum of sum
    derive Find a positive number that the sum of the number and its inverted value was minimal.
  • 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. Find the container's radius r (and height h) so that they can hide the largest possible treasure.
  • 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 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.
  • Rectangle pool
    basen Find dimensions of an open pool with a square bottom with a capacity of 32 m3 to have painted/bricked walls with the least amount of material.
  • Cube surfce2volume
    cube_shield Calculate the volume of the cube if its surface is 150 cm2.
  • Find the
    eq222 Find the number x, which if it increases by 2, then its square increases by 21 percent.
  • 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 Find the surface of the cube with a volume of 27 dm3.
  • Completing square
    eq2 Solve the quadratic equation: m2=4m+20 using completing the square method
  • Two gardens
    garden The flower garden has a square shape. The new garden has the shape of a rectangle, and one dimension is 8 m smaller, and the other is twice as large as in a square garden. What were the original garden dimensions and the new garden if both gardens' area i
  • 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)
  • The cube
    cubes The cube has a surface area of 486 m ^ 2. Calculate its volume.
  • Area to volume
    two_cubes If the surface area of a cube is 486, find its volume.
  • Two numbers and its product
    ratios The product of two numbers are 2/3. If on of them is 1/10, what is the other?