The following generalization of Ostrowski's inequality is given: Let \(f\in C^{n+1}([a,b])\), \(n\in\mathbb{N}\) and \(y\in [a,b]\) be fixed, such that \(f^{(k)}(y)=0\), \(k=1,\dots,n\). Then \[ \Biggl|{1\over b-a} \int^b_a f(t)dt- f(y)\Biggr|\leq {|f^{(n+1)}|_\infty\over (n+2)!} \Biggl({(y-a)^{n+2}+ (b-y)^{n+2}\over b-a}\Biggr). \] The inequality is sharp. Namely, when \(n\) is odd it is attained by \(f^*(z):= (z-y)^{n+1}\cdot(b-a)\), while when \(n\) is even the optimal function is \(\overline f(z):=|z-y|^{n+\alpha}\cdot(b-a)\), \(\alpha>1\).