Mathematics

Problem

A differentiable function

g

has tangent-line approximations

L_{4}

and

L_{6}

constructed at

x=4

and

x=6

. For points near

x=4

L_{4}

consistently gives values less than

g

, while for points near

x=6

L_{6}

gives values greater than

g

. What must occur at least once on the open interval

(4,6)

regarding

g''(x)

? Explain your conclusion in one sentence.