Is #f(x)=xe^(x^3-x)-1/x^3# increasing or decreasing at #x=4#?

1 Answer

#f(x)# is INCREASING at #x=4#

Explanation:

the given equation is #f(x)=x*e^(x^3-x)-1/x^3#

Let us solve first the first derivative then substitute the value #x=4#

If #f' (4)=#positive value, then it is INCREASING at #x=4#

If #f' (4)=#negative value, then it is DECREASING at #x=4#

If #f' (4)=#zero value, then it is STATIONARY at #x=4#

First derivative

#f' (x)=x*d/dx(e^(x^3-x))+e^(x^3-x)*d/dx(x)-d/dx(x^(-3))#

#f' (x)=x*(e^(x^3-x))*d/dx(x^3-x)+e^(x^3-x)*1-(-3)(x^(-3-1))#

#f' (x)=x*(e^(x^3-x))*(3x^2-1)+e^(x^3-x)+3(x^(-4))#

Evaluate at #x=4#

#f' (4)=4*(e^(4^3-4))*(3(4)^2-1)+e^(4^3-4)+3(4^(-4))#

#f' (4)=4*(e^(4^3-4))*(3(4)^2-1)+e^(4^3-4)+3(4^(-4))#

#f' (4)="positive value"#

Therefore INCREASING at #x=4#

God bless....I hope the explanation is useful.