Respuesta :
Answer:
-5
Step-by-step explanation:
Given [tex]f(x)=3x^{5}+5x^{4}[/tex].
Derivative of [tex]f(x)[/tex] is given by [tex]\dfrac{df(x)}{dx} =15x^{4}+20x^{3}=g(x)[/tex].
Minimum value of [tex]g(x)[/tex] can be found differentiating again and equating it to zero.
[tex]\dfrac{dg(x)}{dx}=60x^{3}+60x^{2}=0\\\\x^{2}=0\text{ (or) }60x+60=0\\x=0\text{ (or) }x=-1[/tex]
Function [tex]g(x)[/tex] takes a minimum value when [tex]\dfrac{d^{2}g(x)}{dx^{2}}>0[/tex]
For, [tex]g''(x)=180x^{2}+120x[/tex], [tex]g''(0)=0;\\g''(-1)=60>0[/tex]
So, minimum value occurs at [tex]x=-1[/tex];
Minimum value = [tex]15(-1)^{4}+20(-1)^{3}=-5[/tex]
∴ Minimum value = -5
