Question

If \(f(x)=a x^{3}+b\), find \(a\) and \(b\) if it is known that \(f(1)=1\) and \(f(2)=15\).

Short answer

The cubic function satisfying the given conditions is \(f(x) = 2x^3 - 1\), where \(a = 2\) and \(b = -1\).

Step-by-step solution

Substituting the given values in the function

Using the given points \((1,1)\) and \((2,15)\), substitute these values into the equation \(f(x) = ax^3 + b\). For the first point \(f(1) = 1\): \[ 1 = a(1)^3 + b \] For the second point \(f(2) = 15\): \[ 15 = a(2)^3 + b \] Now we have a system of linear equations: \[ \begin{cases} 1 = a+b \\ 15 = 8a+b \end{cases} \]

Solve the linear system of equations

Next, we'll solve for either a or b using one of the equations, then substitute that value into the other equation to solve for the remaining variable. From the first equation, we can write: \[ b = 1 - a \] Now substitute the expression for b into the second equation: \[ 15 = 8a + (1 - a) \]

Solve for a

Now, solve for a: \[ 15 = 8a + 1 - a \implies 15 - 1 = 7a \implies a = 2 \]

Substitute and solve for b

Substitute the value of a back into the expression for b: \[ b = 1 - a \implies b = 1 - 2 \implies b = -1 \]

Write the final cubic function

Now that we have the values for a and b, we can write the final cubic function as: \[ f(x) = 2x^3 - 1 \]