Find the intervals where it is increasing, decreasing, concave up, and concave down, local extreme values, and inflection points.
y=e^x-1-x
I took the derivative and got, e^(x-1) -1+e^(x-1)-x
what to do next?
If $\displaystyle y = e^{(x-1)} - x$,
Then $\displaystyle \frac{dy}{dx} = \frac{d}{dx} (x-1) * e^{(x-1)} - 1$
and $\displaystyle \frac{d}{dx} (x-1) = 1$
So $\displaystyle \frac{dy}{dx} = e^{(x-1)} - 1$
To solve for local extreme values (maxima and minima), solve $\displaystyle \frac{dy}{dx} = 0$
To solve for where the function is increasing, solve $\displaystyle \frac{dy}{dx} > 0$
To solve for where the function is decreasing, solve $\displaystyle \frac{dy}{dx} < 0$
For concave up and concave down the second derivative is needed.
$\displaystyle \frac{d^2 y}{dx^2} = \frac{d}{dx} . \frac{dy}{dx}$
$\displaystyle \frac{d^2 y}{dx^2} = e^{(x-1)}$
To find where the function is concave up, solve $\displaystyle \frac{d^2 y}{dx^2} > 0$
To find where the function is concave down, solve $\displaystyle \frac{d^2 y}{dx^2} < 0$
To find where the inflection points, solve $\displaystyle \frac{d^2 y}{dx^2} = 0$