Determine a polynomial based on values of its derivatives

by RoRi

September 18, 2015

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,

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

Stumbling Robot