Maths helpppp pleaseeee

Stormiii

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the sum of two numbers is 64. show that their product has a maximum of 1024.
 

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the sum of two numbers is 64. show that their product has a maximum of 1024.
You don't want to find two numbers whose product is 1024, but to show that the product is never greater than 1024.

That is, show that P = x(64-x) has a peak at 1024. What is the vertex of this parabola?
 
For all readers, the graph of

y = ax^2 + bx + c

is a parabola that opens upwards or downwards, and the x-coordinate of the parabola's vertex point is given by

x = -b/(2a)




Hint for remembering: For those who've learned about the Quadratic Formula and its Discriminant, you can remember -b/(2a) as the x-value we get from the Quadratic Formula when the Discriminant is zero.

?
 
For all readers, the graph of

y = ax^2 + bx + c

is a parabola that opens upwards or downwards, and the x-coordinate of the parabola's vertex point is given by

x = -b/(2a)




Hint for remembering: For those who've learned about the Quadratic Formula and its Discriminant, you can remember -b/(2a) as the x-value we get from the Quadratic Formula when the Discriminant is zero.

?
thank you!!!!
 
For all readers, the graph of

y = ax^2 + bx + c

is a parabola that opens upwards or downwards, and the x-coordinate of the parabola's vertex point is given by

x = -b/(2a)




Hint for remembering: For those who've learned about the Quadratic Formula and its Discriminant, you can remember -b/(2a) as the x-value we get from the Quadratic Formula when the Discriminant is zero.

?
thank u!!!
have you studied the equation of parabola and associated quadratic equations ?
yep, don't worry I've figured it just today, thank u!
 
To wrap up:

Let the numbers be 'a' & 'b '. Then

a + b = 64 .....so........ b = 64 - a

a * b = a *(64 - a) = -a^2 + 64 * a = P(roduct)

Maximum value of P @ a =- (-64)/[2*(-1)] = 32 ..... & b = 64 - 32 = 32

Check

a + b = 64 ...... & ... a * b = 32 * 32 = 1024..... Checks
 
It can be shown that given a sum, that the max product will be half the sum squared.
So the max product will be (.5*64)^2 = 32^2 =1024
 
Last edited:
To wrap up:

Let the numbers be 'a' & 'b '. Then

a + b = 64 .....so........ b = 64 - a

a * b = a *(64 - a) = -a^2 + 64 * a = P(roduct)

Maximum value of P @ a =- (-64)/[2*(-1)] = 32 ..... & b = 64 - 32 = 32

Check

a + b = 64 ...... & ... a * b = 32 * 32 = 1024..... Checks
thank uuu
 
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