Question

Find the area of the triangle formed from the coordinate axes and the tangent line to the curve

Answer 1

tangent line to the curve at point (x₀, y₀):

$y-y_0=f'(x_0)(x-x_0)$

intersections of tangent line with axes:

$y=0\implies -y_0=f'(x_0)(x-x_0)$

$x=-y_0/f'(x_0)+x_0$

$x=0\implies y=-f'(x_0)x_0+y_0$

So, the area of the triangle:

$S=|xy|/2=|(-y_0/f'(x_0)+x_0)(-f'(x_0)x_0+y_0)|/2$