Doubly Tangent Line

Let

$L(x) = m x + b$

be the linear function whose graph is tangent to $f(x)$ at two distinct points. Define

$g(x) = f(x) - L(x)$

Then, since $f(x)$ opens upward, it follows that, $f(x)$ has to be above $L(x)$ for all $x$ except at the points of tangency, where they are coincident. This means $g(x)$ is always non-negative.

Now,

$g(x) = x^4 - 8 x^3 + 17 x^2 + (2 - m) x + (-24 - b)$

Since $g(x)$ is non-negative and has two roots, then these two roots must be double roots, and therefore,

$g(x) = ( x^2 + c x + d )^2$

for specific constants $c, d$ .

Expanding,

$g(x) = x^4 + 2 c x^3 + (2d + c^2 ) x^2 + 2 c d x + d^2$

Comparing the two forms of $g(x)$ , we deduce that

$c = -4$ , $d = 0.5$ . Hence, $m = 6$ and $b = -24.25$

and, therefore, $m + b = -18.25$