An elementary problem on calculus. I was going through the chapter on "The Principle Of Least Action" of The Feynman Lectures, and came across a property of minimum for a well-behaved function dependent on a single-variable (as I have deduced by merely looking at the figures given in the book), which he puts forth thus:
"If we go away from the minimum in the first-order, the deviation of the function from its minimum value, is only second-order."
After a few lines, he again states:
"At any place else on the curve, if we move a small distance, the value of the function changes in the first-order also."
Using basic concepts from calculus, I tried to prove these two statements on my own. I succeeded in proving the latter, but got stuck on the former.
Here's how I proceeded:
I assume a twice-derivable(of course, a special case, but simple to analyse) function $f(x)$, dependent on a single variable $x$, and emphasize, that within a certain interval,f(x) is well-behaved. I don't need to know its behavior out of that interval.
At any general point,
$f'(x)=k_1(x)-------(1);  $$ f"(x)=k_2 $(independent of x)$-------(2); k,c$ belongs to the set of reals.
 so, starting from equn.. (2), I get:
$f(x)=(k_2/2)(x^2) + ax + b$, where, a,b are arbitrary constants, 
and, $k_1(x)=cx + a$
let $(x_1,f(x_1)), (x_2,f(x_2))$ be two co-ordinates of interest. Then,  $f(x_1)=(k_2/2)(x_1^2) + ax_1 + b$, and,  $f(x_2)=(k_2/2)(x_2^2) + ax_2 + b$,  hence, $f(x_2) - f(x_1) = (c/2) [(x_2)^2 - (x_1)^2] + a(x_2 - x_1)$,, or,  change in $f(x)$= linear combination of second and first-order changes in $x$,  which duly proves the second part.
What I find troublesome is to prove the first part, as well as the second part for a general function (dependent on single-variable only). As you have noticed, I have assumed $f(x)$ to be well-behaved, "twice"-derivable function of $x$.

At a minimum point in a function, $f'(x) = 0$ but $f''(x) > 0$.  The fact that there is a nonzero second derivative means that the dominant (lowest-order) term in $f$ at that point is from the $x^2$.
For the second statement, any function can be approximated as a straight-line tangent to the function.  This is the first-order term in the Taylor expansion:
$$f(x) = f(a) + f'(a)(x-a) + ...$$
