Polynomial interpolation with data points from derivative of original polynomial

Question

Question: What is a polynomial g(x) no more than degree 3 (including 3) s.t. $$g(0) = 1, g(1) = 0, g′(0) = 0, g′(−1) = −1$$

Solution is: $$g(x) = −\frac3 5x^3 − \frac2 5x^2 + 1$$

My attempt: I was thinking of setting up the general cubic which has the form $g(x) = ax^3 + bx^2 + cx + d$. Then plugging in the four points given in the function will give me four equations in the four unknowns $\{a,b,c,d\}$. But there is derivatives here. (I barely learned about lagrange interpolation, does that have anythign to do with this problem?)

Answer 1

We have

$$g(x) = ax^3 + bx^2 + c x + d \implies g'(x) = 3ax^2 + 2bx + c$$

We also have

$$g(0) = a (0) + b(0) + c(0) + d = 1 \implies d = 1$$

Can you continue?

Find $g(1), g'(0), g'(-1)$ and solve for the last three constants.