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Determine a polynomial based on values of its derivatives

Let P(x) = ax^3 + bx^2 + cx + d such that

    \[ P(0) = P(1) = -2, \qquad P'(0) = -1, \qquad P''(0) = 10. \]

Find the values of the constants a,b,c,d.


First, let’s compute the first and second derivatives,

    \begin{align*}  && P(x) &= ax^3 + bx^2 + cx + d \\ \implies && P'(x) &= 3ax^2 + 2bx + c \\ \implies && P''(x) &= 6ax + 2b. \end{align*}

Now, we start determining values based on the conditions given,

    \[ P''(0) = 10 \quad \implies \quad 2b = 10 \quad \implies \quad b = 5. \]

Next, using this value we have P'(x) = 3ax^2 + 10x + c, so,

    \[ P'(0) = -1 \quad \implies \quad c = -1. \]

Then, plugging these values into the expression for P we have P(x) = ax^3 +5x^2 -x + d. So,

    \[ P(0) = -2 \quad \implies \quad d = -2, \]

and,

    \[ P(1) = -2  \quad \implies \quad a + 5 - 1 -2 = -2 \quad \implies \quad a = -4. \]

Therefore,

    \[ a = -4, \quad b = 5, \quad c = -1, \quad d = -2. \]

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