114. a) Prove the inequality $\sqrt{n+1}+2 \sqrt{n}<\sqrt{9 n+3}$ for all natural numbers n; b) Does there exist a natural number n such that $[\sqrt{n+1}+2 \sqrt{n}]<[\sqrt{9 n+3}]$, where $[a]$ denotes the integer part of the number a?

115. Find the maximum and minimum values of the function $y=\cos x(\cos x+1)(\cos x+2)(\cos x+3)$.
114. a) Prove the inequality $\sqrt{n+1}+2 \sqrt{n}<\sqrt{9 n+3}$ for all natural numbers $n$; b) Does there exist a natural number $n$ for which $[\sqrt{n+1}+2 \sqrt{n}]<[\sqrt{9 n+3}]$, where $[a]$ denotes the integer part of the number $a$?

Answer: b) No such $n$ exists. Hint: See problem 10.5.

115. Find the maximum and minimum values of the function
$y=\cos x(\cos x+1)(\cos x+2)(\cos x+3)$.

Answer: The minimum value is -1, and the maximum value is 24. Hint: Let $u=\cos x, u \in[-1 ; 1]$. Then $y=f(u)=u(u+1)(u+2)(u+3)=(u(u+3))((u+1)(u+2))=$ $\left(u^{2}+3 u\right)\left(u^{2}+3 u+2\right)$. Let $t=u^{2}+3 u$. Note that the vertex of the quadratic function $t(u)$ has the coordinate $u_{0}=-3 / 2$ and therefore $t(u)$ is monotonically increasing on the interval $[-1 ; 1]$, taking values from -2 to 4. Further, the function $y(t)=t(t+2)=(t+1)^{2}-1$ takes its minimum value of -1 at $t=-1$, and at the endpoints of the interval $[-2 ; 4]$ it takes the values 0 and 24, respectively, from which the result follows.