Find a function $f(x, y, z)$ such that $\nabla f$ is the constant vector $\langle 9,2,6\rangle$. (Use symbolic notation and fractions where needed. Use $C$ for the constant of integration.)

Step 1 ：We are given that the gradient of the function \(f\), denoted as \(\nabla f\), is the constant vector \(\langle 9,2,6\rangle\). This means that the partial derivatives of \(f\) with respect to \(x\), \(y\), and \(z\) are 9, 2, and 6, respectively.

Step 2 ：From the partial derivative with respect to \(x\), we can write \(f(x, y, z) = 9x + g(y, z)\) for some function \(g(y, z)\).

Step 3 ：Similarly, from the partial derivative with respect to \(y\), we can write \(f(x, y, z) = 2y + h(x, z)\) for some function \(h(x, z)\).

Step 4 ：Comparing these two expressions for \(f\), we get \(9x + g(y, z) = 2y + h(x, z)\). This implies that \(g(y, z) - h(x, z) = 2y - 9x\).

Step 5 ：From the partial derivative with respect to \(z\), we can write \(f(x, y, z) = 6z + i(x, y)\) for some function \(i(x, y)\).

Step 6 ：Comparing this with our previous expression for \(f\), we get \(6z + i(x, y) = 2y - 9x + g(y, z)\). This implies that \(i(x, y) - g(y, z) = 2y - 9x - 6z\).

Step 7 ：From these, we can conclude that the function \(f(x, y, z)\) is of the form \(9x + 2y + 6z + C\) for some constant \(C\).

Step 8 ：Final Answer: The function is \(f(x, y, z) = \boxed{9x + 2y + 6z + C}\).