We show that Durr-Hoyer's quantum algorithm of searching for extreme point of integer function can not be sped up for functions chosen randomly. Any other algorithm acting in substantially shorter time o(√(2^n)) gives incorrect answer for the functions with the single point of maximum chosen randomly with probability converging to 1. The lower bound as Ω (√(2^n /b)) was established for the quantum search for solution of equations f(x)=1 where f is a Boolean function with b such solutions chosen at random with probability converging to 1.