Question

The normal to the curve y= x^3 + cx at the point (2,d) has gradient 1/2. Find the values of c and d and hence the equation of the normal.

Answer 1

The normal line to this curve at (2, d) has slope 1/2, so the tangent line to this curve at (2, d) has slope -2.

y = x³ + cx

y' = 3x² + c

3(2²) + c = -2

12 + c = -2

c = -14

y = x³ - 14x

d = 2³ - 14(2) = 8 - 28 = -20