For what values of a,b,and c do the graphs off (x)=x^2+ax+b and g(x)=x^3+cx have the same tangent line at the point (2, 2)?

Answer 1

Since the point (2,2) is on both graphs we have f(2) = 2 and g(2) = 2.

So, 4 + 2a + b = 2 and 8 + 2c = 2. So, c = -3.

f'(x) = 2x + a and g'(x) = 3x² + c = 3x² - 3

Since f'(2) = g'(2), we have 4 + a = 9. So, a = 5.

Since 4 + 2a + b = 2 and a = 5, we see that b = -12.

a = 5, b = -12, and c = -3