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When you want to locate the location for the maximum rate of change of a single variable function, then one method is to use the second derivative test(f''=0).

In the multivariable case we can use the gradient. I know that the gradient vector points in the direction of where the function's rate of change is maximum but I was unable to solve my problem unless I was given the hint that I should check maximum of $\left | \bigtriangledown f \right |^{2}$. I was asked to determine at what location the steepest point is. I cannot find an argument in my textbook about this so I ask here: What does $\left | \bigtriangledown f \right |^{2}$ describe?

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$|\nabla f|^2$ is the squared magnitude of the gradient. The gradient vector describes the direction of greatest increase at any given point. Finding the maximum magnitude of this vector tells you where the increase is greatest (or more precisely, greatest in magnitude).

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Well since $\left | \triangledown f \right |$ describes the direction of the greatest increase, then I should maximize $\left | \triangledown f \right |$? Is not that right? Why squared? –  EricAm Jan 29 '13 at 20:30
    
$\nabla f$ is a vector, and so it describes the direction of greatest increase. $|\nabla f|$ is that vector's magnitude, defined to be positive, so there is no difference between maximizing it and maximizing its square. –  Muphrid Jan 29 '13 at 20:39

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