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Ask iit jee aieee pet cbse icse state board community Community Discussion Question: Quadratic Expressions
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sachin_gupta1991 (69)

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Q. Let f(x)= ax2 + bx + c, where a,b,c belongs to R. Suppose that mod.{f(x)}<=1 for all x belongs to [0,1].
Find the greatest values of mod{a},mod{b} and mod{c}.
    
pramod6990 (945)

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is the answer 8, 8 and1...
this is one method.....
for it to be a limiting case let us concider an eqn. in whos graph, f(1)=1, f(0)=1 and  consequently f(1/2)= -1
substituting it in the eqn.s for ax2+bx+c we get 3 eqn.s solving which we have
a=8,b=-8 and c=1
moda=8, modb=8, mod c=1
 
or
 
we can imagine of another eqn. in which we have f(1)=-1, f(0)= -1 and consequently f(0)=1
following the same procedure for this eqn. we obtain a= -8 b=8 and c= -1
but again moda=8 modb=8 and modc=1
 
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"Logic is the systematic way of reaching the wrong conclusion with confidence" lol.....
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sachin_gupta1991 (69)

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Your answers for mod{a},mod{c} are correct(8 and 1) but that of mod{b} is incorrect.
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pramod6990 (945)

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then what is the correct answer.....
im eager to know about.....
not worried abt the correct answer  yaar i jus wanna know whether my funda applied is correct or not......silly mistake bhi ho sakta hai but what i want to know is whether the method was right.....cud u post the soln in detail if u have it???? if no then plz invite some one else like hsbhatt sir to come and have a look at this one.....

"Logic is the systematic way of reaching the wrong conclusion with confidence" lol.....
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hsbhatt (4450)

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I had seen a question that went:
	ext {Let}  f(x) = ax^2+bx+c  	ext{with a,b,c}  in R \ \

	ext{If}  |f(x)| leq1  	ext{for all}  x in (0,1) \ \

	ext {Then prove that}  |a|+|b|+|c|leq17
 
The solution goes:
 
|f(1)|=|a+b+c|leq 1; |f(rac{1}{2})|=|rac{a}{4}+rac{b}{2}+c|leq 1; |f(0)|=|c|leq 1
Then |a|=2|f(1)-2f(rac12)+f(0)|leq 2|f(1)|+4|f(rac12)|+2|f(0)|leq 8
|b|=|f(1)-4f(rac12)+3f(0)|leq |f(1)|+4|f(rac12)|+3|f(0)|leq 8 

Time wounds all heels
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