Mathematics : Application of Derivatives

Find points on the curve  at which the tangents are (i) parallel to the x-axis (ii) parallel to the y-axis.

Find points on the curve  at which the tangents are (i) parallel to the x-axis (ii) parallel to the y-axis.

For the curve y = 4x³ – 2x⁵, find all the points at which the tangent passes through the origin.

Show that the curves 2x = y² and 2xy = k cut at right angles if k² = 8

Show that the curve xy = a² and x² + y² = 2a² touch each other.

If the curve αx² + βy² = 1 and α' x² + β'y² = 1 intersect orthogonally, prove that (α – α') β β') = (β – β') α α'.