Consider the function and the tangent to the curve at
Show that the tangent does not pass through the origin if .
It seems obvious that it does that it pass through the origin when I draw the function for as the origin is 'inside' the parabola. But is this argument enough?
I also have the equation of the tangent being
An alternative solution:
At x = a, the tangent line goes through the point and the slope is . Therefore, writing the equation of the line in slope-intercept form,
, where b is the y-intercept.
. If b = 0, then we get , which cannot be true since k < 0 and the LHS is always non-negative.