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## 'A Function of Gradient' printed from http://nrich.maths.org/

A continuous, differentiable curve $y = f(x)$ enters a region of
the x-y plane with gradient $+1$ at the point $(1, 1)$ and emerges
from the region with gradient $+1$ at either point $A=(3, 4)$,
$B=(3, 3)$ or $C=(3, 2)$. Another curve $y = g(x)$ enters a region
of the x-y plane with gradient $+1$ at the point $(1, 3)$ and
emerges from the region with gradient $-1$ at either point $D=(3,
4)$, $E=(3, 3)$ or $F=(3, 2)$.

For each of the 6 possibilities, sketch

- no equations required -
possible forms of the graphs of the gradients of the functions
against $x$.

In which cases must the gradient function necessarily be zero at at
least one place in the obscured regions?

In which cases must the gradient of the gradient function
necessarily be zero at at least one place in the obscured regions?
Give a clear argument.

Find equations for 6 functions which have the correct gradients at
these points.

Extensions: If the gradients at points
$A$ to $F$ were allowed to vary, under which circumstances might
there be no zeros in the gradients and the gradients of the
gradients? Explore the possibility of two different functions
matching the conditions for $B$.

NOTES AND BACKGROUND

The mathematical ideas involved in this question give some insight
into a university-level analytical way of viewing calculus. For
example, a key result proved in first year analysis courses is
the Mean Value Theorem, which states that for functions which are
differentiable for $a< x< b$ we can always find a number
$c$ such that $f ^{\, '}(c)=\frac{f(b)-f(a)}{b-a}$.