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Let f(x) = a x^2 + bx + c, where a, b, c...

Let `f(x) = a x^2 + bx + c`, where `a, b, c in R, a!=0`. Suppose `|f(x)| leq1, x in [0,1]`, then

A

`|a|le8`

B

`|b|le8`

C

`|c|le1`

D

`|a|+|b|+|c|le17`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
We have `f(x)=ax^(2)+bx+c`
`a,b,c epsionR[ :' a!=0]`
On puttig `x=0,1,1/2`, we get
`|c|le1`
`|a+b+c|le1`
and `|1/4a+1/2b+c|le1`
`implies-1lecle1`,
`-1lea+b+cle1`
and `-4lea+2b+4cle4`
`implies-4le4a+4b+4cle4`
and `-4le-a-2b-4cle4`
On adding we get
`-8le3a+2ble8`
Also `-8lea+2ble8`
`:.-16le2ale16`
`implies|a|le8`
`:'-1le-cle1,-8le-ale8`
We get `-16le2ble16`
`implies|b|le8`
`:.|a|+||+|c|le17`
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