The number of mosquitoes, M(x) in millions of Dhaka city depends on rainy seasons' rainfall, x, in inches approximately as: M(x) = 12x - x². What would be the maximum of mosquitoes in Dhaka city?

Question: The number of mosquitoes, M(x) in millions of Dhaka city depends on rainy seasons' rainfall, x, in inches approximately as: M(x) = 12x - x². What would be the maximum of mosquitoes in Dhaka city?

Solution:
দেওয়া আছে, M(x) = 12x - x²
বা, M(x) = - (x² - 12x)
বা, M(x) = - (x² - 2 . x . 6 + 6² - 36)
বা, M(x) = - {(x - 6)² - 36}
∴ M(x) = 36 - (x - 6)²

সর্বোচ্চ মানের জন্য (x - 6)² = 0 হতে হবে।
∴ M(x)_max = 36 - 0 = 36

নির্ণেয় মশার সর্বোচ্চ সংখ্যা 36 মিলিয়ন।

বিকল্প সমাধান:
Solution:
The equation is M(x) = - x² + 12x, where a = - 1 and b = 12.
For a quadratic function, f(x) = ax² + bx + c:
the maximum occurs at x = - b/2a.
∴ সর্বোচ্চ মানের জন্য, x = - b/2a
= - 12/{2(- 1)} = 6

x = 6 হলে,
M(6) = 12(6) - (6)²
= 72 - 36
= 36

∴ The maximum number of mosquitoes is 36 million.