The directional derivative of the scalar function f(x, y, z) = x2 + 2y2 + z at the point P = (1, 1, 2) in the direction of the vector \[\overrightarrow {\rm{a}} = 3{\rm{\hat i}} – 4{\rm{\hat j}}\] is A. -4 B. -2 C. -1 D. 1

The directional derivative of a scalar function $f$ at a point $P$ in the direction of a vector $\mathbf{a}$ is given by:

$$D_{\mathbf{a}}f(P) = \nabla f(P) \cdot \mathbf{a}$$

where $\nabla f$ is the gradient of $f$.

In this case, we have $f(x, y, z) = x^2 + 2y^2 + z$ and $\mathbf{a} = 3\hat{\imath} – 4\hat{\jmath}$. Therefore, the directional derivative is:

$$D_{\mathbf{a}}f(P) = \nabla f(P) \cdot \mathbf{a} = (2x + 0y + 1z)(3\hat{\imath} – 4\hat{\jmath}) = 3 – 4 = -1$$

Therefore, the correct answer is $\boxed{-1}$.

To explain each option in brief:

• Option A is incorrect because $D_{\mathbf{a}}f(P) = -1$, not $-4$.
• Option B is incorrect because $D_{\mathbf{a}}f(P) = -1$, not $-2$.
• Option C is incorrect because $D_{\mathbf{a}}f(P) = -1$, not $-1$.
• Option D is correct because $D_{\mathbf{a}}f(P) = -1$.