Question

Help with num 9 please. thanks​

Answer 1

See Below.

Explanation:

We want to show that the function:

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

Increases for all values of x.

A function is increasing whenever its derivative is positive.

So, find the derivative of our function:

`\displaystyle f'(x) = (d)/(dx)\left[e^x - e^(-x)\right]`

Differentiate:

`\displaystyle f'(x) = e^x - (-e^(-x))`

Simplify:

`f'(x) = e^x+e^(-x)`

Since eˣ is always greater than zero and e⁻ˣ is also always greater than zero, f'(x) is always positive. Hence, the original function increases for all values of x.