1950 AHSME Problems/Problem 41

Problem

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

$\textbf{(A)}\ -\dfrac{b}{a} \qquad \textbf{(B)}\ -\dfrac{b}{2a} \qquad \textbf{(C)}\ b^2-4ac \qquad \textbf{(D)}\ \dfrac{4ac-b^2}{4a}\qquad \textbf{(E)}\ \text{None of these}$

Solution

The vertex of a parabola is at $x=-\frac{b}{2a}$ for $ax^2+bx+c$ . Because $a>0$ , the vertex is a minimum. Therefore $a(-\frac{b}{2a})^2+b(-\frac{b}{2a})+c=a(\frac{b^2}{4a^2})-\frac{b^2}{2a}+c=\frac{b^2}{4a}-\frac{2b^2}{4a}+c=-\frac{b^2}{4a}+c=\frac{4ac}{4a}-\frac {b^2}{4a}=\frac{4ac-b^2}{4a} \Rightarrow \mathrm{(D)}$ .

Solution 2

Using calculus, the derivative of the function is $2ax + b$ , which has a critical point at $\frac{-b}{2a}$ . The second derivative of the function is $2a$ ; since $a > 0$ , this critical point is a minimum. As in Solution 1, plug this value into the function to obtain $\boxed{\textbf{(D)}\ \dfrac{4ac-b^2}{4a}}$ .

1950 AHSC (Problems • Answer Key • Resources)
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1950 AHSME Problems/Problem 41

Contents

Problem

Solution

Solution 2

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