11sin^2 a + 9cos^2 a + 8sin^4 a + 2cos^4 a = = 9sin^2 a + 9cos^2 a + 2sin^2 a + 6sin^4 a + 2(sin^4 a + 2cos^4 a) = (*) Заметим, что 1) 9sin^2 a + 9cos^2 a = 9(sin^2 a + cos^2 a) = 9 2) sin^4 a + cos^4 a = sin^4 a + 2sin^2 a*cos^2 a + cos^4 a - 2sin^2 a*cos^2 a = = (sin^2 a + cos^2 a)^2 - 2sin^2 a*cos^2 a = 1 - 1/2*(4sin^2 a*cos^2 a) Подставляем (*) = 9 + 2sin^2 a + 6sin^4 a + 2 - 4sin^2 a*cos^2 a = = 11 + 4sin^2 a - 2sin^2 a + 6sin^4 a - 4sin^2 a*cos^2 a = = 11 - 2sin^2 a + 6sin^4 a + 4sin^2 a*(1 - cos^2 a) = = 11 - 2sin^2 a + 6sin^4 a + 4sin^4 a = 11 - 2sin^2 a + 10sin^4 a = = 10(sin^4 a - 2*1/10*sin^2 a + 1/100) - 1/10 + 11 = = 10(sin^2 a - 1/10)^2 + 109/10 Минимальное значение квадрата равно 0, а всего выражения 109/10.
= 9sin^2 a + 9cos^2 a + 2sin^2 a + 6sin^4 a + 2(sin^4 a + 2cos^4 a) = (*)
Заметим, что
1) 9sin^2 a + 9cos^2 a = 9(sin^2 a + cos^2 a) = 9
2) sin^4 a + cos^4 a = sin^4 a + 2sin^2 a*cos^2 a + cos^4 a - 2sin^2 a*cos^2 a =
= (sin^2 a + cos^2 a)^2 - 2sin^2 a*cos^2 a = 1 - 1/2*(4sin^2 a*cos^2 a)
Подставляем
(*) = 9 + 2sin^2 a + 6sin^4 a + 2 - 4sin^2 a*cos^2 a =
= 11 + 4sin^2 a - 2sin^2 a + 6sin^4 a - 4sin^2 a*cos^2 a =
= 11 - 2sin^2 a + 6sin^4 a + 4sin^2 a*(1 - cos^2 a) =
= 11 - 2sin^2 a + 6sin^4 a + 4sin^4 a = 11 - 2sin^2 a + 10sin^4 a =
= 10(sin^4 a - 2*1/10*sin^2 a + 1/100) - 1/10 + 11 =
= 10(sin^2 a - 1/10)^2 + 109/10
Минимальное значение квадрата равно 0, а всего выражения 109/10.
1) f'(x)=(sin²x)'=2*sinx*cosx=sin2x; f''(x)=(sin2x)'=2cos2x;
2)f'(x)=(cos2x)'=-2*sin2x; f''(x)=(-2sin2x)'=-4cos2x
3) f'(x)=(√x)'=1/(2√x)=(x⁻¹/²)/2; f''(x)=((x⁻¹/²)/2)'=-(1/4)*x⁻³/²=-1/(4x√x)
4) f'(x)=(x²-2√x)'=2x-(2/(2√x))=(2x-x⁻¹/²); f''(x)=(2x-(x⁻¹/²))'=2-(-1/2)*x⁻³/²=
2+1/(2x√x);
5) f'(x)= (xsinx)'=sinx+x*cosx; f''(x)=сosx+cosx-x*sinx=2cosx-x*sinx;
6) f'(x)=(xcos3x)'=cos3x-3x*sin3x; f''(x)=(cos3x-3x*sin3x)'=-3sin3x-3sin3x-9x*сos3x=-6sin3x-9x*сos3x;
7) f'(x)=(3x²-cos(x²+1))'=6x+sin(x²+1)*(2x)=2x*(3+sin(x²+1)); f''(x)=
(2x*(3+sin(x²+1)))'=
2*(3+sin(x²+1))+2x*(2x*cos(x²+1))=6+2sin(x²+1)+4x²*cos(x²+1);
8) f'(x)=(sin²2x)'=2*sin2x*(cos2x)*2=2sin4x; f''(x)=(2sin4x)'=8*cos4x;
9) f'(x)=(x²sin2x)'=2x*sin2x+2x²*cos2x;
f''(x)=(2x*sin2x+2x²*cos2x)'=2*sin2x+4x*cos2x+4x*cos2x-4x²*sin2x=
2*sin2x+8x*cos2x-4x²*sin2x