Sup-norm bound $\max|f|\le\frac18\max|f''|$ with zero boundary values

Let \(f\) be a twice continuously differentiable function on \([0,1]\) such that \(f(0)=f(1)=0\). Prove that \[ \max_{x\in[0,1]}|f(x)|\le \frac18\max_{x\in[0,1]}|f''(x)|, \] and find an example where equality holds.