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For any point, in the first quadrant, on the line 4x + 9y = 36, what is the maximum value of x6 * y6 ?
  • a)
    39
  • b)
    312
  • c)
    310
  • d)
    311
Correct answer is option 'B'. Can you explain this answer?
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For any point, in the first quadrant, on the line 4x + 9y = 36, what i...
4x + 9y = 36
Both x and y are positive in the first quadrant ⇒ AM ≥ GM inequality holds true.
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For any point, in the first quadrant, on the line 4x + 9y = 36, what i...
Problem:
For any point in the first quadrant on the line 4x + 9y = 36, what is the maximum value of x^6 * y^6?

Solution:
To find the maximum value of x^6 * y^6, we need to find the maximum value of x and y on the given line. Let's solve the equation 4x + 9y = 36 for y in terms of x.

4x + 9y = 36
9y = 36 - 4x
y = (36 - 4x)/9

Maximizing x:
To maximize x, we need to find the x-coordinate of the point where the line intersects the x-axis. When y = 0, we have:

y = (36 - 4x)/9
0 = (36 - 4x)/9
36 - 4x = 0
4x = 36
x = 9

So, the maximum value of x on the line is 9.

Maximizing y:
To maximize y, we need to find the y-coordinate of the point where the line intersects the y-axis. When x = 0, we have:

y = (36 - 4x)/9
y = (36 - 4(0))/9
y = 36/9
y = 4

So, the maximum value of y on the line is 4.

Calculating x^6 * y^6:
Now that we have the maximum values of x and y, let's calculate x^6 * y^6:

x^6 = 9^6 = 531441
y^6 = 4^6 = 4096

x^6 * y^6 = 531441 * 4096 = 2176782336

Therefore, the maximum value of x^6 * y^6 is 2176782336, which corresponds to option B.
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For any point, in the first quadrant, on the line 4x + 9y = 36, what is the maximum value of x6 * y6?a)39b)312c)310d)311Correct answer is option 'B'. Can you explain this answer?
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