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Couple of months ago I had an entrance exam wherein this problem appeared. (I hope this is what it was).

For a scalar function [tex]f\left(x\right)=f\left(x_{1},x_{2},...,x_{n}\right)[/tex] the gradient is given as

[tex]\nabla f=\left(\frac {\partial f \left(x\right)} {\partial x_1},\frac {\partial f \left(x\right)} {\partial x_2},...,\frac {\partial f \left(x\right)} {\partial x_n}\right)[/tex]

Then show that for any small change [tex]\Delta x[/tex], [tex]f\left(x+\Delta x\right)[/tex] is maximum if [tex]\Delta x[/tex] lies along [tex]\nabla f\left(x\right)[/tex].

Frankly, I did not get the question then. So I did few preliminary steps as follows.

[tex]f\left(x+\Delta x\right)=f\left(x\right)+\nabla f\left(x\right)\cdot\Delta x + O\left(\left(\Delta x\right)^2\right)[/tex]

Hence,

[tex]\nabla f\left(x+\Delta x\right)=\nabla f\left(x\right)+\nabla \left(\nabla f\left(x\right)\cdot\Delta x\right) [/tex]

For [tex]f\left(x+\Delta x\right)[/tex] to be maximum, [tex]\nabla f\left(x+\Delta x\right)=0[/tex] is maximum

Hence,

[tex]\nabla f\left(x\right)=-\nabla \left(\nabla f\left(x\right)\cdot\Delta x\right) [/tex]

This is where I gave up then. Bu intuition I could see that the dot product would give a maximum answer if [tex]\Delta x[/tex] lies along [tex]\nabla f\left(x\right)[/tex]. But I could not prove it.

So can somebody help me with this as this question is haunting me for last two months now.

Thanks in advance.

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# Gradient Ascent question

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