If I am asked to determine the parameters of a quadratic function
$f(x) = a\cdot x^2 + b\cdot x + c$
given the information that
$f(1) = 1$
$f(3) = 5$
$f'(1) = 0$
I can easily determine the parameters a, b and c by setting up a system of linear equations
$a + b + c = 1$
$9\cdot a + 3\cdot b + c = 5$
$2\cdot a + b = 0$
which then gives me the equation
$f(x) = x^2 - 2\cdot x + 2$.
My question is now: How can one incorporate the constraint that the slope of the function f(x) is not allowed to exceed a value L for a range [x1, x2] i.e. $max(f'(x)) <= L, ~~\forall x \in [x1, x2]$?
This is NOT just about the example above but a general question; how would one write down this constraint in mathematical terms and use it to determine the parameters? Any suggestions?
To find possible maxima of the derivative, we can look at the zeros of the second derivative. For the quadratic function that you describe $f(x)=ax^2+bx+c$, this is especially easy since: \begin{equation} f''(x)=2a \neq 0 \qquad \text{if} \qquad a\neq0. \end{equation} This means that you will never have an interior maximum of the slope (as you can see also from the graph of a parabola). The maximum is thus attained at one of its boundary points. The constraint can be written as \begin{equation} \max{\{f'(x_1),f'(x_2)\}}\leq L. \end{equation} So, you can now solve the problem as above and then check for the solution if it matches the constraint, by just evaluating the two boundary points.
For more general functions, it might also be nice to get an expression for the points where $f''(x)=0$. If you find these points in terms of your parameters, say $(x_3(a,b,c),x_4(a,b,c),\ldots)$, you would have to check the constraint for these points and the boundary points: \begin{equation} \max{\{f'(x_1),f'(x_2),f'(x_3),f'(x_4),\ldots\}}\leq L. \end{equation}
The above is just a way of checking if the found solution is good, but you could of course use your expressions for the interior maxima to specify the domain in which you match the constraint. For example, you could get constraints like \begin{equation} \text{lower}(a,b,c)\leq x \leq \text{upper}(a,b,c) \end{equation}