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Question

Let f(x)=ax2+bx+c where a,b, and c are constants. If f(x) takes it maximum value at x=13, then which of the following is necessarily true?


A

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B

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C

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D

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Solution

The correct option is A


f(x)=ax2+bx+c
Since maxima of f(x) occurs at x=13,f(x) can be rewritten as f(x)=a(x−13)2+k (where f(13)=k)
It is obvious that a< 0.
Assuming f(n)=f(0) where n is any constant.
f(0)=9(19)+k=f(n)=a(n−13)2+k
⇒(n−13)2=19
⇒n=23
Therefore,f(0)=f(23)

Alternative Method:
The maximum value of f(x)=ax2+bx+c occurs at x=−b2a.
−b2a=13⇒b=−23a
Also,f(0)=c. Substituting b=−23ain f(x),we can easily observe that f(0)=f(23).


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