1)(x+y)^3-(x-y)^3-2y = (x + y - x + y) * ((x + y)² + (x + y)(x - y) + (x - y)²) - 2y =
= 2y * (x² + 2xy + y² + x² - y² + x² - 2xy + y²) - 2y = 2y * (3x² + y²) - 2y = 2y(3x² + y² - 1)
2) (m+n)^3+(m-n)^3-2m = m³ + 3m²n + 3mn² + n³ + m³ - 3m²n + 3mn² - n³ - 2m =
= 2m³ + 6mn² - 2m = 2m( m² + 3n²) - 2m = 2m * (m² + 3n² - 1)
3) (a-b)^3-(c+d)^3-a+b+c+d = (a - b - c -d) ((a-b)² + (a - b)(c + d) + (c + d)²)- a + b + c + d =
= (a - b - c -d) ((a-b)² + (a - b)(c + d) + (c + d)²)- (a - b - c - d) = (a - b - c - d) ((a+ b)((a +b + c + d) + + (c + d)² - 1 )
![\left(\frac{\sqrt[m]{x}+\sqrt[m]{y}}{2}\right)^m\le \left(\frac{\sqrt[n]{x}+\sqrt[n]{y}}{2}\right)^n,](/tpl/images/4512/3019/e49ae.png)
2-2cos^2(x)-5cosx-5=0
2cos^2(x)+5cosx+3=0
cosx=t, -1<=t<=1
2t^2+5t+3=0
D=25-24=1
t=(-5+-1)/4
-1<=t<=1
t=-1
cosx=-1
x=п+пn, n - целое число
2) 4(1-sin^2(x))-3sinx-3=0
4-4sin^2(x)-3sinx-3=0
4sin^2(x)+3sinx-1=0
sinx=t, -1<=t<=1
4t^2+3t-1=0
D=9+16=25
t=(-3+-5)/8
-1<=t<=1
t=-1
t=1/4
sinx=-1
sinx=1/4
x=-п/2+2пn, n - целое число
x=arcsin1/4+2пk, k - целое число
х=п-arcsin1/4+2пl, l - целое число
3) 2sin((x+3x)/2)sin((x-3x)/2)=0
sin2x=0
sin(-x)=0
sin2x=0
sinx=0
2x=пn, n - целое число
х=пk, k - целое число
х=пn/2
4) 2sin((3x+x)/2)cos((3x-x)/2)=0
sin2x=0
cosx=0
2x=пn, n - целое число
х=п/2+пk, k - целое число
х=пn/2
1)(x+y)^3-(x-y)^3-2y = (x + y - x + y) * ((x + y)² + (x + y)(x - y) + (x - y)²) - 2y =
= 2y * (x² + 2xy + y² + x² - y² + x² - 2xy + y²) - 2y = 2y * (3x² + y²) - 2y = 2y(3x² + y² - 1)
2) (m+n)^3+(m-n)^3-2m = m³ + 3m²n + 3mn² + n³ + m³ - 3m²n + 3mn² - n³ - 2m =
= 2m³ + 6mn² - 2m = 2m( m² + 3n²) - 2m = 2m * (m² + 3n² - 1)
3) (a-b)^3-(c+d)^3-a+b+c+d = (a - b - c -d) ((a-b)² + (a - b)(c + d) + (c + d)²)- a + b + c + d =
= (a - b - c -d) ((a-b)² + (a - b)(c + d) + (c + d)²)- (a - b - c - d) = (a - b - c - d) ((a+ b)((a +b + c + d) + + (c + d)² - 1 )