The x-intercept of the line tangent y=ax^3-3x+b to at (2,3) is 1. Find the values of a and b.

Answer 1

(2, 3) is point on the given and we have: 3 = a·2³ - 3·2 + b, or 8a + b = 9.

Slope of tangent line: y' = 3ax² - 3; slope m = 3a·2² - 3 = 12a - 3.

Equation of tangent line: y - 3 = (12a -3)(x - 2); x- intercept is (1,0), becouse we have 0 - 3 = (12a - 3)(1 - 2);

12a - 3 = 3; 12a = 6; a = 6/12 = 1/2. Then 8·1/2 + b = 9; b = 5.

Answer: a = 1/2, b = 5