A calculus problem by Sayantan Dhar
If 2 a + 3 b + 6 c = 0 , which of the following domains must contain a root to the equation
a x 2 + b x + c = 0 ?
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1 solution
If a root is between 0 and 1, it is also between -1 and 1. So 2 options are the answer...
I was talking about the problem itself, with a = b = 1 and c = -5/6 I got roots which doesn't fit any of the options.
Are you sure? Take a = b = 1, c = -5/6 has roots ~ -1.54 and 0.54
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Bro I am saying -1,1 also includes 0,1 i.e. if answer is 0,1 it is in -1,1 too. Haha
well, in the question it is stated shortest range , so only 0,1 is the answer
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You changed the problem definition after I posted the comment.
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actually i wanted to mean that from the beginning but my english is a bit poor
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@Sayantan Dhar – Haha. I was saying that for others. Anyways those who understand shortest range will get right haha
@Sayantan Dhar – No problem!
Nice question and solution. :)
It is very easy , if you consider Rolle's Theorem .
Let f ′ ( x ) = a x 2 + b x + c f ( x ) = 3 a x 3 + 2 b x 2 + c x + C f ( x ) = 6 1 ( 2 a x 3 + 3 b x 2 + 6 c x + 6 C ) f ( 0 ) = C , f ( 1 ) = 6 1 ( 2 a + 3 b + 6 c + 6 C ) = C
Which implies that a root must lie between ( 0 , 1 )
Consider the equation 2a x 3 + 3b x 2 +6cx=0 The roots of this equation are 0 and 1. so clearly its derived equation will have a root between 0 and 1 since its graph will take a turn between these values. its derived equation is 6a x 2 + 6bx + 6c=0 or a x 2 + bx + c=0 so the equation a x 2 +bx +c =0 will have a root between 0 and 1