Multiple Choice QuestionsThe point on the curve x2 = 2y which is nearest to the point (0, 5) is
(0, 0)
(0, 0)
Short Answer TypeDetermine two positive numbers whose sum is 15 and the sum of whose, squares is minimum.
Amongst all pairs of positive numbers with product 64, find those whose sum is the least.
Here x + y = 40 ⇒ y = 40 – x ...(1)
Let f (x) = x y3 = x (40 – x)3
f'(x) =Â 
= x [3(40 – x)2 (–1)] + (40 – x)3 Â
= – 3 x (40 – x)2 +(40 – x) 3
= (40 – x)2 (– 3 a + 40 – x) = (40 – x)2 (– 4 x + 40)
= 4 (40 – x)2 (10 – x)
f ' (x) = 0 ⇒  4 (40 – x)2 (10 – y) = 0 ⇒ x = 10, 40
Rejecting a = 40 as 0 < x < 40, we get, x = 10
When x < 10 slightly, f ' (x) = 4 (+) (+) = +ve
When x > 10 slightly, f ' (x) = 4 (+)(–) = –ve
∴ at x = 10, f ' (x) changes from +ve to –ve
∴ f (x) has local maximum at x = 10
But f (x) has only one extreme point x = 10
∴ f (x) is maximum when x = 10, y = 40 – 10 = 30
∴ x = 10, y = 30.
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Find two positive numbers a and y such that their sum is 35 and the product x2y5 is a maximum.
Find two positive numbers a and y such that their sum is 35 and the product x2y5 is a maximum.
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