Step-by-step explanation:
Step 1: Take the First Derivative This means only differentiate once.
Disclaimer: Since absolute value only take positve outputs and quadratics only take positve outputs, we can get rid of the absolute value signs so we now have
![e {}^{ {x}^{2} - 1 }](https://tex.z-dn.net/?f=e%20%7B%7D%5E%7B%20%7Bx%7D%5E%7B2%7D%20%20-%20%201%20%7D%20)
We have the function x^2-1 composed into the function e^x.
So we use chain rule
Which states the derivative of a function composed is the
derivative of the main function times the derivative of the inside function.
So the derivative of the main function is
![\frac{d}{dx} (e {}^{x} ) = e {}^{x}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28e%20%7B%7D%5E%7Bx%7D%20%29%20%3D%20e%20%7B%7D%5E%7Bx%7D%20)
Then we replace x with x^2-1
![e {}^{ {x}^{2} - 1}](https://tex.z-dn.net/?f=e%20%7B%7D%5E%7B%20%7Bx%7D%5E%7B2%7D%20%20-%201%7D%20)
Then we take the derivative of the second function which is 2x so qe multiply them
![e { }^{ {x}^{2} - 1 } 2x](https://tex.z-dn.net/?f=e%20%7B%20%7D%5E%7B%20%7Bx%7D%5E%7B2%7D%20-%201%20%7D%202x)
Step 2: Set the equation equal to zero.
![e {}^{x {}^{2} - 1} 2x = 0](https://tex.z-dn.net/?f=e%20%7B%7D%5E%7Bx%20%7B%7D%5E%7B2%7D%20-%201%7D%202x%20%3D%200)
Since e doesn't reach zero. We can just set 2x=0.
![2x = 0 = x = 0](https://tex.z-dn.net/?f=2x%20%3D%200%20%3D%20%20x%20%3D%200)
So the critical point is 0.
Since e^x will never reach zero
Since 0 is the only critical point, this where the max or min will occur at.
Next we pick any numbergreater than zero, and plug them in the derivative function which gives us a positve number.
Any pick less than zero will give us a negative number.
Since the function is decreasing then increasing, we have a minimum.
Since 0 is the only critical point, we have a absolute minimum at 0.
To find the y coordinate, plug 0 in the orginal function.
Which gives us
![e {}^{ {0}^{2} - 1 } = e {}^{ - 1} = \frac{1}{e}](https://tex.z-dn.net/?f=e%20%7B%7D%5E%7B%20%7B0%7D%5E%7B2%7D%20-%201%20%7D%20%20%3D%20e%20%7B%7D%5E%7B%20-%201%7D%20%20%3D%20%20%5Cfrac%7B1%7D%7Be%7D%20)
So the minimum occurs at
(0,1/e).