Question 1181254
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Points A and B determine the line with equation y=2x+1.<br>
Here is a graph of that line and the given quadratic:<br>
{{{graph(400,400,-5,5,-1,10,2x+1,x^2+4x+7)}}}<br>
Using AB as the base of the triangle, with P somewhere on the parabola, the minimum area of triangle PAB will be when the height of the triangle is smallest; that will happen when P is at the point on the parabola which is closest to the line.  And that will be where the tangent to the parabola has the same slope as the line.<br>
y=x^2+4x+7
y'=2x+4<br>
The slope of the line is 2; find the point on the parabola where the slope of the tangent is 2:<br>
2x+4=2
2x=-2
x=-1
y=1-4+7=4<br>
The area of PAB is minimum when P is (-1,4).<br>
Then use the "shoelace" method to find the area of the triangle with vertices (0,1), (2,5), and (-1,4).<br><pre>

      0   1
        X
      2   5
    /   X   \
  2  -1   4   0
    /   X   \
 -5   0   1   8
    /       \
  0          -1
 ---         ---
 -3           7

Area = (1/2)(7-(-3)) = 5<br>
ANSWER: The minimum area of PAB is 5.<br>