If \(f(x) = 3x^{3} + 8x^{2} + 6x + k\) and \(f(2) = 1\), find the value of k.
Answer Details
The function \(f(x)\) is given as \(f(x) = 3x^3 + 8x^2 + 6x + k\), and we are given that \(f(2) = 1\). To find the value of \(k\), we can substitute \(x = 2\) and \(f(2) = 1\) into the function and solve for \(k\).
\begin{align*}
f(2) &= 3(2)^3 + 8(2)^2 + 6(2) + k \\
1 &= 24 + 32 + 12 + k \\
1 &= 68 + k \\
k &= 1 - 68 \\
k &= -67
\end{align*}
Therefore, the value of \(k\) is -67.