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Let f be a function of three (real) variables having continuous partial derivatives. For each direction vector $h = (h_1, h_2, h_3)$ such that $h_1^2+ h_2^2+ h_3^2=1$, let $ D_hf (x, y, z)$ be the directional derivative of $f$ along $h$ at $(x, y, z)$. For a point $(x_0, y_0, z_0)$ where the partial derivative $∂/∂xf (x_0, y_0, z_0)$ is not zero, maximize $D_hf (x_0, y_0, z_0)$ (as a function of h) Totally stuck. Can somebody help how should I solve the problem? |
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Here's an expansion of my hint in the comment above. Step 1: Have you already learned about the gradient $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$? If not, prove that for any vector $v$, the directional derivative of $f$ in the direction $v$ is $\nabla f \cdot v$. Step 2: For what unit vector $\hat{h}$ is $\nabla f\cdot \hat{h}$ maximized? Hint: $\nabla f \cdot \hat{h} = \|\nabla f\|\cos \theta$, where $\theta$ is the angle between $\nabla f$ and $\hat{h}$. |
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