Question

Решите неравенство f'(x)<0
1) f(x)= 3/x-x^2
2) f(x)=x^4-4,5x^2+2
3) f(x)=-1/x^2-1/x-5
4)f(x)=x^4-4x-3

Answer 1

$1) quad f(x) = frac{3}{x} - x^2\ \ f'(x) = -frac{3}{x^2}-2x; quad f'(x) < 0\ \ -frac{3}{x^2}-2x < 0 \ \ frac{3}{x^2}+2x>0 \ \ frac{3 +2x^3}{x^2} > 0 \ \ \ ----(sqrt[3]{-frac 3 2})+++(0)++++>_x \ \ x in (sqrt[3]{-frac 3 2};0)cup (0; +infty)$

$2) quad f(x) = x^4 - 4.5 x^2+2 \ \ f'(x) = 4x^3-9x;\ \ 4x^3 - 9x = 0 \ \ x(4x^2 - 9) = 0 \ \ x (2x-3)(2x+3) = 0 \ x_1 = 0, ~x_2 = frac 3 2, ~ x_3 = -frac 3 2 \ \ f'(x) < 0 \ \ ---(-frac 3 2)+++(0)---(frac 3 2)+++>_x \ \ x in (-infty;-frac 3 2) cup(0; frac 3 2)$

$3) quad f(x) = -frac{1}{x^2} -frac{1}{x} -5\ \ f'(x) = frac{2}{x^3}+frac{1}{x^2}; \ \ f'(x) < 0 \ \ dfrac{2+x}{x^3} < 0 \ \ +++(-2)---(0)+++>_x \ \ x in (-2; 0)$

$4) quad f(x) = x^4 -4x-3 \ \ f'(x) = 4x^3-4 \ \ f'(x) < 0 \ \ 4x^3 - 4 < 0 quad Big |:4 \ \ x^3 -1 < 0 \ \ ---(1)+++>_x\ \ xin (-infty; 1)$