Problem: The least value of the function ax2+bx+c(a>0) is:
Answer Choices:
A. −ab​
B. −2ab​
C. b2−4ac
D. 4a4ac−b2​
E. none of the above.
Solution:
The graph of the function y≡ax2+bx+c,aî€ =0, is a parabola with its axis of symmetry (the line MV extended) parallel to the y-axis.
We wish to find the coordinates of V. Let y=k be any line parallel to the x-axis which intersects the parabola at two points, say P and Q. Then the abscissa (the x-coordinate) of the midpoint of the line segment PQ is the abscissa of V. Since P and Q are the intersections of the line y=k with the parabola, the abscissas of P and Q must satisfy the equation
ax2+bx+c=k or ax2+bx+c−k=0
Its roots are:
x=−2ab​+2a1​​Dˉ and x′=−2ab​−2a1​​Dˉ,
where D=b2−4ac is the discriminant.
∴ the abscissa of V=2x+x′​=−2ab​, and the
ordinate of V=a(−.2ab​)2+b(−2ab​)+c=−4ab2​+c=4a4ac−b2​
if a>0, the ordinate of V is the minimum value of the function ax2+bx+c and if a<0, it is the maximum value.
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