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[tex]\frac{1}{x}\int_{0}^{x}f (t) \,dt \le \int_{0}^{1}f (t) \,dt [/tex]

So by working backwards I got to trying to show that [itex](1-x)\int_{0}^{1}f (t) \,dt \le \int_{x}^{1}f (t) \,dt [/itex]. While I know both sides are equal at x=1, the LHS is decreasing, and the RHS is continuous, I haven't been able to find a reason to say the inequality is true (probably nothing there).

Should I pass to considering upper and lower sums, or is there some fancy way to use integration by parts here? I haven't come across anything compelling in all my random scratch work.