question_answer 1)

Find points at which are tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis. 

Answer:

Let P(h, k) be the required point.        lies on given curve                    y = x3 ? 3x2 ? 9x + 7                  ?. (1)       K = h3 ? 3h2 ? 9h + 7                  ?. (2)       Diff. (1) w.r.t. x             m = slope of tangent to (1)                   Now the tangent to (1) is || to X-axis if m =0        3h2 ? 6h ? 9 = 0        h2 ? 2h ? 3 = 0        h2 ? 2h + h ? 3 = 0        h (h ? 3) + (h ? 3) = 0        (h ? 3) (h + 1) = 0        h = 3, ?1       when h = 3, (2)        K = 27 ? 27 ? 27 + 7 = ?20       when h = ?1, (2)        K = ?1 ? 3 + 9 + 7 = 12        Required points are (3, ?20), (?1, 12).