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I have had an answer which differentiates the function and equates it to zero.

But why?

How does the differential bring us to find the "closest" point on a function to another point?

I thought that maybe because it was to the origin we are taking away the general point of the function from the origin (0,0) and then differentiating this to find the minimum but that doesn't work when you do the algebra?

This question is in the middle of finding the closest point on the line $y= x+1$ to the origin by the way

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    You're on the right track. Write "distance from a point on the graph to the origin" as a function and find its minimum. – 2017-01-11
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    The procedure you describe would find the point at minimum distance from the $x$-axis – 2017-01-11
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    Is the question correct or is this already in the middle of finding the point closest to the origin of the line $y=x+1$? – 2017-01-11
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    @LutzL It's in the middle – 2017-01-11
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    @LutzL which is actually important I should mention it! Thanks – 2017-01-11
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    OK so the question is about finding the closest point of graph with points $(x,x+1)$ to the origin, so minimize $f(x)=x^2+(x+1)^2$, so solving $f'(x)=0$? – 2017-01-11
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    @Maestro13 the way you worded it makes more sense but now I wanna know how you turned $(x, x+1)$ to that function? – 2017-01-11
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    $d((0,0),(x,x+1))=\sqrt{(x-0)^2+(x+1-0)^2}$ and minimizing $g(x)=\sqrt{x^2+(x+1)^2}$ will give the same result as minimizing $f(x)=x^2+(x+1)^2$. – 2017-01-12

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You want to minimize the norm of $(x,2x^2+2x+1)$, this is the same as minimizing $x^2+(2x^2+2x+1)^2$. The latter is a polynomial of degree $4$ that takes on non-negative values, so to find the critical points you must solve a degree $3$ polynomial.

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    I don't understand this? Sorry? – 2017-01-11
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    do you know how to minimize a polynomial? you find all of the points were the derivative is zero and plug them in. The derivative of that polynomial is a degree $3$ polynomial. – 2017-01-11
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    Yeah but I don't get how the derivative will minimize it specifically to the point (0,0), what if I wanted the closest point to the point (1,1) instead of the origin? – 2017-01-11
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    then the function that you need to minimize is different, although still a polynomial – 2017-01-11