Question

Solve the following inequalities.
$f\left(x\right)⩾g^{\prime }\left(x\right)$, if $f\left(x\right)=10x^3-13x^2+7x,g\left(x\right)=11x^3-15x^2-3$

Answer 1

Given that $f\left(x\right)=10x^3-13x^2+7x$

and $g\left(x\right)=11x^3-15x^2-3$

the given condition is $f\left(x\right)\ge g^{\prime }\left(x\right)$

so $g\left(x\right)=11x^3-15x^2-3$

$g^{\prime }\left(x\right)=33x^2-30x$

now put the value

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

$10x^3-13x^2+7x\ge 33x^2-30x$

$10x^3-46x^2+37x\ge 0$

$10x^2-46x+37\ge 0$

by using $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ find the roots

$\frac{46\pm \sqrt{{46}^2-4\times 10\times 37}}{2\times 10}$

$\frac{46\pm \sqrt{636}}{20}$

$=\frac{46+25.2190}{20}$$=3.46095$

$=\frac{46-\sqrt{636}}{20}$$=1.0390$

so $x\in (1.0390,3.46095)$