Question

Solve the following inequality:
$f^{\prime }\left(x\right)⩾g^{\prime }\left(x\right)$, if $f\left(x\right)=5-3x+\frac{5}{2}{x}^2-\frac{x^3}{3},g\left(x\right)=3x-7$

Answer 1

$f^{{}^{\prime }}\left(x\right)\ge g\left(x\right)$

$\frac{d}{dx}(0-3x+\frac{5}{2}{x}^2-\frac{x^3}{3})\ge 3x-7$

$0-3x+5x-3x^2>3x-7$

$-3+2x-3x^2>-7$

$-3+2x-3x^2+7>0$

${3x}^2-2x-4<0$

${3x}^2-6x-2x-4<0$

$3x(x-2)+2(x-2)<0$

$(3x+2)(x-2)$$<0$

$xϵ(-\frac{2}{3},2)$