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Suppose f(x) is a function satisfying th...

Suppose `f(x)` is a function satisfying the following conditions: `f(0)=2,f(1)=1` `f` has a minimum value at `x=5/2` For all `x ,f^(prime)(x)=|2a x2a x-1 2a x+b+1bb+1-1 2(a x+b)2a x+2b+1 2a x+b|` where `a , b` are some constants. Determine the constants `a , b` , and the function `f(x)`

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The correct Answer is:
` a = (1)/(4), b = (-5)/(4); f(x) = (1)/(4) x ^(2) - ( 5)/(4) x + 2 `
Given, ` f' (x) = |{:(2ax,, 2ax - 1 ,, 2ax + b + 1), (b,, b + 1,, - 1 ), (2(ax+ b ),, 2ax+ 2b + 1,, 2ax + b ):}|`
Applying ` R_3 to R_3 - R_1 - 2 R_2`, we get
`f ' (x) = |{:(2ax,, 2ax- 1,, 2ax+ b + 1 ), (b,, b + 1,,- 1),(0,, 0,, 1):}|`
`rArr f' (x) = 2ax + b `
On integrating both sides, we get
`" " f(x) = a x ^(2) + b x + c `
Since, maximum at ` x = 5 //2 rArr f ' (5//2) = 0`
`rArr " " 5a + b = 0 " " `... (i)
Also, ` " " f (0) = 2 rArr c = 2 " " `... (ii)
and `" " f (1) = 1 rArr a + b + c = 1 " "`...(iii)
On solving Eqs. (i), (ii) and (iii), we get
`" " a = (1)/(4), b = - (5)/(4), c = 2 `
Thus, ` " " f(x) (1)/(4) x ^(2) - ( 5)/(4) x + 2 `
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