The cubic function is shown on the coordinate plane (graph not provided)
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
The cubic function h(x)=x3−6x2+9x is shown on the coordinate plane (graph not provided). A family of tangent lines to h is considered.
(a) Derive h′(x).
(b) The line y=−3x+5 is given. Find all x–values where a tangent to h is parallel to this line.
(c) For each x found in part (b), write the corresponding tangent line in general (standard) form Ax+By+C=0 with integer coefficients and gcd(A,B,C)=1.
(d) Indicate which, if any, of these tangent lines pass through the origin. Justify your conclusion algebraically.