Maximum value of a function with 2 variables. If x and y are real numbers such that x^2-10x+y^2+16=0, determine the maximum value of the ratio y/x

Paula Cameron

Paula Cameron

Answered question

2022-11-04

Maximum value of a function with 2 variables
Can someone help me finding maximum value of a ratio in quadratic function in 2 variables using proper mathematical methods.?
Question is as below.
If x and y are real numbers such that x 2 10 x + y 2 + 16 = 0, determine the maximum value of the ratio y/x
I know there is Ramban method to solve this. Taking y / x = k > y = k x and forming equation in x, then applying 2 4 a c >= 0 for max min value of k.
Is there any way to using differentiation ?

Answer & Explanation

artirw9f

artirw9f

Beginner2022-11-05Added 20 answers

Step 1
y / x = m .
x 2 10 x + ( m x ) 2 + 16 = 0.
( 1 + m 2 ) x 2 10 x + 16 = 0.
( 1 + m 2 ) ( x 2 10 1 + m 2 x ) + 16 = 0.
Completing the square:
( 1 + m 2 ) ( x 5 1 + m 2 ) 2 25 1 + m 2 + 16 = 0.
( 1 + m 2 ) 2 ( x 5 1 + m 2 ) 2 = 16 ( 1 + m 2 ) + 25 0.
Step 2
Hence:
25 / 16 1 + m 2
9 / 16 m 2
Maximal m:
m = 3 / 4.

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