There exists one and only one straight line which is tangent to the curve
𝑦 = 𝑥^4 − 9𝑥^2 at two points. determine the x coordinates of these two points.
2 answers
so would (-3,0) and (3,0) count or would that be considered as a secant line?
no, those two points are where the graph crosses the x-axis. One has a negative slope and the other a positive. No way could a single line be tangent at both of them.
y' = 4x^3 - 18x = 2x(2x^2-9)
y'=0 at x = ±3/√2
Since y is an even function, y(a) = y(-a) so the horizontal line through (-3/√2,-81/4) and (3/√2,-81/4) is tangent at both points.
Not only that, you can be sure it is the only such point, since y' is an odd function, so y'(-a) = -y'(a) and there are no other pairs of points with the same value.
y' = 4x^3 - 18x = 2x(2x^2-9)
y'=0 at x = ±3/√2
Since y is an even function, y(a) = y(-a) so the horizontal line through (-3/√2,-81/4) and (3/√2,-81/4) is tangent at both points.
Not only that, you can be sure it is the only such point, since y' is an odd function, so y'(-a) = -y'(a) and there are no other pairs of points with the same value.