1950 AMC 12 Problem 41

Attempt Problem 41 of the 1950 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1950 AMC 12 solutions, or check the answer key.

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41.

The least value of the function ax2+bx+cax^2+bx+c with a>0a>0 is:

ba-\dfrac ba

b2a-\dfrac{b}{2a}

b24acb^2-4ac

4acb24a\dfrac{4ac-b^2}{4a}

None of these

Answer: D
Concepts:completing the squarequadraticoptimization
Difficulty rating: 1580
Small Hint:

Complete the square in ax2+bx+cax^2+bx+c

Big Hint:

Since a>0,a>0, the squared term is minimized when it equals zero

Solution:

Completing the square gives ax2+bx+c=a(x+b2a)2+cb24a. \begin{aligned} ax^2+bx+c &=a\left(x+\frac{b}{2a}\right)^2\\ &\quad+c-\frac{b^2}{4a}. \end{aligned} Because a>0,a>0, the squared term has minimum 0.0. The least value is therefore cb24a=4acb24a. c-\frac{b^2}{4a}=\frac{4ac-b^2}{4a}.

Thus, the correct answer is D.

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Problem 41 in Other Years

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