213а)
f(x) =2x² -x.
f'(x) =4x -1.
f'(x) =0;
4x-1 =0;
x=1/4.
213б)
f(x) =2/3x³ +x² +12.
f'(x) =(2/3x³ +x² +12)=2/3*3x² +2x +0 =2x² +2x.
f'(x) =2x² +2x.
f'(x) =0;
2x² +2x=0;
2x(x+1)=0 ;
x₁ = 0;
x+1 =0;
x₂ = - 1.
Сделал подробно г213г) f(x) =x³/3-1,5x² -4x .
f '(x) = (x³/3-1,5x² -4x) ' =(x³/3) ' -(1,5x²) ' - (4x) = 1/3(x³)' -1,5(x²)' -4(x)' =
1/3*3x² -1,5*2x - 4*1 =x² -3x -4.
f '(x) =x² -3x -4.
f '(x) =0 ;
x² -3x -4. =0
x₁ = -1;
x₂= 4.
213д)
f(x) =2x -5x².
f'(x) =(2x -5x²) ' =2 -10x.
f'(x) =0;
2-10x=0;
10x=2 ;
x=1/5 ( =0,2).
***********************************************************
214б)
f(x) =x³ +1,5x² .
f ' (x) =(x³ +1,5x²) ' =(x³)' +(1,5x²)' =3x² +1,5(x²)' =3x² +1,5*2x =3x² +3x.
f ' (x) =3x² +3x.
f '(x) <0 ;<br>3x² +3x < 0;
3x(x+1) <0;<br>x(x+1)<0;<br> + - +
---------------- -1 --------- 0 -------------
x ∈ ( -1; 0) .
214г)
f(x) =4x -1/3*x³ .
f ' (x) =(4x -1/3*x³) ' =(4x) ' -(1/3*x³) ' =4(x)' -1/3(x³)'
=4*1 -1/3*3x² =4 -x² .
f ' (x) =4 -x² .
f'(x) < 0;
4 -x² <0 ;<br>x² -4 >0;
(x+2)(x-2) > 0;
+ - +
---------------- -2 --------------- 2 -------------------
x∈ (-∞; -2) U (2 ;∞).
214a)
f(x)=4x- 3x².
f'(x) =4 -6x;
f'(x) < 0;<br>4 -6x <0 ;<br>6x -4> 0;
x >2/3.
x∈(2/√3 ; ∞).
214в)
f(x)=x² -5x.
f'(x) =2x -5 ;
f'(x) < 0;<br>2x -5 < 0;<br>2x < 5 ;
x < 2,5.
x∈( -∞; 2,5) .