Choice 3 of 4: ∇ f ( P ) ∇f(P) is orthogonal to the level surface f ( x , y , z )

We are given a function $f(x, y, z)$ with the condition that $\nabla f(P) \neq 0$, meaning the gradient of $f$ at the point $P$ is non-zero. We are tasked with identifying which of the given statements about the gradient $\nabla f(P)$ is false. Let's examine each option.

Option 1:

$\nabla f(P)$ indicates the direction of fastest ascent at $P$:

• This is true. The gradient $\nabla f(P)$ points in the direction of the steepest (fastest) increase in the function's value at the point $P$.

Option 2:

$|\nabla f(P)|$ is the maximum value of $D_u f(P)$:

• This is true. The directional derivative $D_u f(P)$ measures the rate of change of $f$ in the direction of the unit vector $u$. The maximum value of this directional derivative occurs when $u$ points in the direction of $\nabla f(P)$, and the magnitude of the gradient $|\nabla f(P)|$ gives this maximum rate of change.

Option 3:

$\nabla f(P)$ is orthogonal to the level surface $f(x, y, z) = 0$:

• This is true. The gradient of a scalar function is always perpendicular (orthogonal) to the level surface defined by the function. In this case, $\nabla f(P)$ is orthogonal to the surface $f(x, y, z) = 0$ at $P$.

Option 4:

All of the above are true:

• This is true, as all the individual statements have been verified to be correct.

Conclusion:

None of the statements are false, so the answer is Option 4: All of the above are true.

Would you like more details or have any questions about this?

Here are some follow-up questions that can expand on this topic:

1. How is the gradient calculated for a multivariable function?
2. What is the geometric interpretation of a level surface?
3. Can the gradient be zero at certain points, and what does that imply?
4. How does the gradient relate to optimization problems?
5. What happens when $\nabla f(P)$ is parallel to a given vector $u$?

Tip: The magnitude of the gradient not only gives the maximum rate of change but also gives information about how quickly the function is changing at a given point.