Question

Is `f(x)=x-e^(2x)-1/x^2` increasing or decreasing at `x=2`?

Answer 1

Decreasing...

Explanation:

To understand if the function is increasing or not, we need to take the differential, as this tells us our gradient, and hence if it's increasing, decreasing etc...

`f(x) = x - e^(2x) - x^(-2)`

`=> f'(x) = d/(dx) ( x - e^(2x) - x^(-2) )`

Using our differential rules...

`d/(dx) ( e^(g(x) )) = g'(x) * e^(g(x))`

`d/(dx) ( x^n ) = nx^(n-1)`

`=> f'(x) = 1 - 2e^(2x) +2x^(-3)`

At `x=2` the differential is `f'(2)`

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

`f'(2) approx -107.946`

Hence at `x=2` the differential is negative, and hence is `underline("decreasing"`