Question

Help with num 9 please. thanks​

Answer 1

Answer:

See Below.

Step-by-step 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:

$\displaystyle f'(x) = \frac{d}{dx}\left[e^x - e^{-x}\right]$

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.