![image](https://tex.z-dn.net/?f=a%29%5C+f%28x%29%3Dx%5E3%2Bx%5C%5C%0Af%27%28x%29%3D%28x%5E3%2Bx%29%27%3Dx%5E2%2B1%3E0%3B%5C+x%5C+E%5C+R%5C%5C%0Af%27%28x%29%3E0%3B%5C+x%5C+E%5C+R%5C%5C%0Ab%29%5C+f%28x%29%3D5x-cosx%5C%5C%0Af%27%28x%29%3D%285x-cosx%29%27%3D5%2Bsinx%3B%5C%5C%0A-1+%5Cleq+sinx+%5Cleq+1%5C%5C%0A4+%5Cleq+5%2Bsinx+%5Cleq+6%5C+%3D%3E%5C+5%2Bsinx%3E0%3B%5C+x%5C+E%5C+R%3B%5C%5C%0Af%27%28x%29%3E0%3B%5C+x%5C+E%5C+R%5C%5C%0Ac%29%5C+f%28x%29%3D1.5x%2Bsinx%5C%5C%0Af%27%28x%29%3D%281.5x%2Bsinx%29%27%3D1.5%2Bcosx%5C%5C%0A-1+%5Cleq+cosx+%5Cleq+1%5C%5C%0A0.5+%5Cleq+1.5%2Bcosx+%5Cleq+2.5%5C+%3D%3E%5C+1.5%2Bcosx%3E0%3B%5C+x%5C+E%5C+Z%5C%5C%0Af%28x%29%3E0%3B%5C+x%5C+E%5C+R%5C%5C%0A%0A)
0;\ x\ E\ R\\
f'(x)>0;\ x\ E\ R\\
b)\ f(x)=5x-cosx\\
f'(x)=(5x-cosx)'=5+sinx;\\
-1 \leq sinx \leq 1\\
4 \leq 5+sinx \leq 6\ =>\ 5+sinx>0;\ x\ E\ R;\\
f'(x)>0;\ x\ E\ R\\
c)\ f(x)=1.5x+sinx\\
f'(x)=(1.5x+sinx)'=1.5+cosx\\
-1 \leq cosx \leq 1\\
0.5 \leq 1.5+cosx \leq 2.5\ =>\ 1.5+cosx>0;\ x\ E\ Z\\
f(x)>0;\ x\ E\ R\\
" alt="a)\ f(x)=x^3+x\\
f'(x)=(x^3+x)'=x^2+1>0;\ x\ E\ R\\
f'(x)>0;\ x\ E\ R\\
b)\ f(x)=5x-cosx\\
f'(x)=(5x-cosx)'=5+sinx;\\
-1 \leq sinx \leq 1\\
4 \leq 5+sinx \leq 6\ =>\ 5+sinx>0;\ x\ E\ R;\\
f'(x)>0;\ x\ E\ R\\
c)\ f(x)=1.5x+sinx\\
f'(x)=(1.5x+sinx)'=1.5+cosx\\
-1 \leq cosx \leq 1\\
0.5 \leq 1.5+cosx \leq 2.5\ =>\ 1.5+cosx>0;\ x\ E\ Z\\
f(x)>0;\ x\ E\ R\\
" align="absmiddle" class="latex-formula">
![image](https://tex.z-dn.net/?f=d%29%5C+f%28x%29%3D3x%2Bcosx-sinx%5C%5C%0Af%27%28x%29%3D%283x%2Bcosx-sinx%29%27%3D3-sinx-cosx%5C%5C%0A-1+%5Cleq+-sinx+%5Cleq+1%5C%5C%0A-1%5Cleq+-cosx%5Cleq+1%5C%5C%0A-1+%5Cleq+-sinx-cosx%5Cleq+1%5C%5C%0A2%5Cleq+3-sinx-cosx%5Cleq5%5C+%3D%3E%5C+3-sinx-cosx%3E0%3B%5C+x%5C+E%5C+R%5C%5C%0Af%27%28x%29%3E0%3B%5C+x%5C+E%5C+R%0A)
\ 3-sinx-cosx>0;\ x\ E\ R\\
f'(x)>0;\ x\ E\ R
" alt="d)\ f(x)=3x+cosx-sinx\\
f'(x)=(3x+cosx-sinx)'=3-sinx-cosx\\
-1 \leq -sinx \leq 1\\
-1\leq -cosx\leq 1\\
-1 \leq -sinx-cosx\leq 1\\
2\leq 3-sinx-cosx\leq5\ =>\ 3-sinx-cosx>0;\ x\ E\ R\\
f'(x)>0;\ x\ E\ R
" align="absmiddle" class="latex-formula">
Пояснение:
Если производная функции больше 0 при любом значении x, то функция возрастает при любом значении x.
x E R = x принадлежит множеству действительных чисел.