# Strictly Convex and Differentiable Implies

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$?

This is true if $f$ is smooth but am wondering if it still holds when $f$ doesn't have a second derivative.

Strong Convexity (Wiki)

• I think it's false. What about linear functions? They're strictly convex but nowhere strongly convex (since the gradient is constant). Perhaps you might want to rule out this scenario though. – bartgol Aug 7 '14 at 22:14
• linear functions aren't strictly convex. Strictly convex: $f(x_2)> f(x_1) +f'(x_1)(x_2-x_1)$ for $x_1 \neq x_2$. – Ashley Aug 7 '14 at 22:19
• My bad. Too tired to think. I'll let someone else jump in. ;-) – bartgol Aug 7 '14 at 22:27

$f(x)=x^4$ is strictly convex, but not strongly convex on any interval including $0$ since its second derivative $12x^2$ won't be bounded above some positive constant near $0.$ [Note this example appeared on the wiki page you cited.]