Question

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

Answer 1

`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.