Is f(x)=x-e^(2x)-1/x^2f(x)=xe2x1x2 increasing or decreasing at x=2x=2?

1 Answer
Feb 10, 2018

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)=xe2xx2

=> 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"