# Answer to Question #14773 in Differential Equations for joanne perez

Question #14773

what is a,b,c and d of the function f(x)=ax^3+bx^2+cx+d with the critical point of (1,2) and (2,3)?

Expert's answer

f'(x) = 3ax² + 2bx + c

f'(1) = 3a + 2b + c = 2& ==> a = 2/3 - 2b/3 - c/3

f'(2) = 3a2² + 2b2 + c = 12a + 4b + c = 3 ==> a = 1/4 - b/3 - c/12

2/3 - 2b/3 - c/3 = 1/4 - b/3 - c/12

5/12 - b/3 - c/4 = 0 ==> b = 5/4 - 3c/4

a = 1/4 - b/3 - c/12 = 1/4 - (5/4 - 3c/4)/3 - c/12 = -1/6 + c/6

So, a = -1/6 + c/6, b = 5/4 - 3c/4, c and d are free parameters.

f'(1) = 3a + 2b + c = 2& ==> a = 2/3 - 2b/3 - c/3

f'(2) = 3a2² + 2b2 + c = 12a + 4b + c = 3 ==> a = 1/4 - b/3 - c/12

2/3 - 2b/3 - c/3 = 1/4 - b/3 - c/12

5/12 - b/3 - c/4 = 0 ==> b = 5/4 - 3c/4

a = 1/4 - b/3 - c/12 = 1/4 - (5/4 - 3c/4)/3 - c/12 = -1/6 + c/6

So, a = -1/6 + c/6, b = 5/4 - 3c/4, c and d are free parameters.

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