Tangent Line to Convex Graph

Theorem
Let $f$ be a real function that is:
 * continuous on some closed interval $\closedint a b$
 * differentiable and convex on the open interval $\openint a b$.

Then all the tangent lines to $f$ are below the graph of $f$.

Proof

 * Concaveup.png

Let $\TT$ be the tangent line to $f$ at some point $\tuple {c, \map f c}$, $c \in \openint a b$.

Let the gradient of $\TT$ be $m$.

Let $\tuple {x_1, y_1}$ be an arbitrary point on $\TT$.

From the point-slope form of a straight line:

For $\TT$:


 * $y = \map \TT x$


 * $y_1 = \map f c$


 * $m = \map {f'} c$


 * $x = x$


 * $x_1 = c$

so:


 * $\map \TT x = \map {f'} c \paren {x - c} + \map f c$

Consider the graph of $f$ to the right of $\tuple {c, \map f c}$, that is, any $x$ in $\openint c b$.

Let $d$ be the directed vertical distance from $\TT$ to the graph of $f$.

That is, if $f$ is above $\TT$ then $d > 0$.

If $f$ is below $\TT$, then $d < 0$.

(From the diagram, it is apparent that $\TT$ is below $f$, but we shall prove it analytically.)

$d$ can be evaluated by:

By the Mean Value Theorem, there exists some constant $k$ in $\openint c b$ such that:

Substitute this into the formula for $d$:

Recall that $x$ lies in the interval $\openint c b$.

So $x > c$, and the quantity $x - c$ is (strictly) positive.

$k$ is also in the interval $\openint c b$ and so $k > c$.

By construction, $f$ is convex.

By the definition of convex:


 * $k > c \implies \map {f'} k > \map {f'} c$

which means that:
 * $\paren {\map {f'} k - \map {f'} c} > 0$

Then $d$ is the product of two (strictly) positive quantities and is itself (strictly) positive.

Similarly, consider the graph of $f$ to the left of $\tuple {c, \map f c}$, that is, any $x$ in $\openint a c$.

By the same process as above, we will have:


 * $d = \paren {\map {f'} k - \map {f'} c} \paren {x - c}$

This time, $x < c$ and the quantity $x - c$ is (strictly) negative.

Further, $k < c$, and so by a similar argument as above:


 * $k < c \implies \map {f'} k < \map {f'} c$

and the quantity $\paren {\map {f'} k - \map {f'} c}$ is also (strictly) negative.

Thus $d$ will be the product of two (strictly) negative quantities, and will again be (strictly) positive.

Also see

 * Tangent Line to Concave Graph
 * Tangent Line at Inflection Point