Question

Find the interval in which the function given $f\left(x\right)=4x^3-6x^2-72x+30$
i) strictly increasing
ii) strictly decreasing

Answer 1

Given,

$f\left(x\right)=4x^3-6x^2-72x+30$

implies $f^{/}\left(x\right)=12x^2-12x-72$

a) for strictly increasing

$f^{/}\left(x\right)>0$

$12x^2-12x-72>0$

$12(x^2-x-6)>0$

$x^2-x-6>0$

$x^2-3x+2x-6>0$

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

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

$x\epsilon (-\infty ,-2)U(3,\infty )$

b) for strictly decreasing

$f^{/}\left(x\right)<0$

$12x^2-12x-72<0$

$12(x^2-x-6)<0$

$x^2-x-6<0$

$x^2-3x+2x-6<0$

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

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

$x\epsilon (-2,3)$