To find: The points on the parabola at which the tangent line cuts from first quadrant the triangle with the smallest area.
The point on the parabola is
Let the points on the curve be .
As the points lie on the parabola write .
Slope of the tangent line is, .
Slope of the tangent line at point it is .
Write the equation of the tangent line by using slope-point form.
Thus, the equation of the tangent line is, .
Find x-intercepts and y-intercepts of the tangent.
To find x-intercept, put .
Thus, the x-intercept is .
Similarly to find y-intercept, put .
Thus, the y-intercept is .
Now area of the triangle formed is given by,
Substitute value of v as .
Differentiate A with respect to u.
Set to get,
Simplify further as follows.
As the point is in first quadrant .
Substitute to get value of v.
Hence, for the values the tangent line to the curve cuts the first quadrant in a triangle with smallest area.
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