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Question
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
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Solution
We have p(x) = 41 − 72x − 18x2
p'(x) = −72 − 36x
Now for critical points, p’(x) = 0
−72 − 36x = 0
−72 = 36x
x = `36/(−72)`
x = −2
p”(x) = −36 < 0
∴ Profit is maximum at x = −2, and maximum profit is p(−2) = 41 − 72 (−2) − 18 (−2)2
= 41 + 144 − 72
= 185 − 72
= 113
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Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
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Hence, rectangle is a square of side `root(5)(2) "cm"`
