You are missing a number of key pieces of information. As someone who has managed people before, let me share them with you.
First of all, when I come to you and tell you I have a problem with your performance, I am doing you a favour. I am giving you a gift. I may not be very good at it, but I am. I could have just stopped by your desk and fired you. Instead, I am trying to help you get better.
Second, if you manage to prove that I said it wrong somehow when I gave you this information, you will not magically cause me to be happy with your performance. That problem will still be there, and in addition, I will have the problem that you think telling me I am wrong is better than solving the problem.
So, "proving" that no-one can know your maximum won't solve your problem and will probably make things worse. What should you do instead? You should ask questions, and listen.
First question. You can't tell from "you are not giving your maximum" whether your boss means:
- you are not producing enough to keep your job. I think you could if you just tried harder, and I really don't want to go through the hassle of firing you and finding someone else, so please try harder or else I will fire you
- you are clearly very smart and capable but you're kind of coasting here because you are so much better than the others. Go ahead and outperform them: I'll reward you when you do.
A good formula for getting more information from a boss and not upsetting them is:
- start with acknowledging what they have said. Make sure your tone and expression match the information. "Oh, I'm sorry to hear you're not pleased with my performance."
- express your position, which should always be that you want the best for your team and your company. "I want to make sure I work to your standards and help the group."
- ask about anything you don't understand. "Are you telling me I'm not meeting a target? Can you tell me more about that?"
As your boss tells you more, don't rebut or argue. Ask more questions. It's ok if some of them are designed to keep things fair ("do you ask everyone to handle that many tasks a week?") as well as just to gain information ("does it matter what project the task is for?"). Don't stop until you understand specifically what you need to start doing differently to have your boss be happy with your performance.
Check in with your boss after about a week and ask "have you seen an improvement in my performance?" You need to know how you're doing. If you're not where you need to be, keep asking things until you understand what is expected of you.