Graphical Interpretation of the derivatives

Question

Just started a new topic, need a little help with the following:

The graph of the function y=x^3+ax^2+bx+c passes through (1,1) and has turning points at x= -1 and x= 3. Find the values of a, b and c

Answer 1

y=x^3+ax^2+bx+c is a polynomial function, Timur. It is continuous and differentiable everywhere.

Plugging in 1, gives 1. This means that

Equation 1 is: a + b + c = 0

y' is 3x^2 + 2ax + b and it equals zero at x = -1, 3

Equation 2 is: 2a - b = 3

Equation 3 is: 6a + b = -27

Solve the three equations for a, b, c and put the results in for the definition of y and you have your polynomial!

y = x^3 -3x^2 -9x + 12

and check it by taking the derivative and plugging in all the values that we were given.