What are the critical values of #f(x)=x/sqrt(x^2+2)-(x-1)^2#?
The critical values of a function can be found by looking at its derivative. If the derivative is either 0 or undefined, this is a critical value.
So, we first start by computing
Let's split this into two derivatives:
We start by computing the first expression:
By the quotient rule, this is equal to
We now have to solve for the derivative of
Now let's substitute back in for
So now we know that
And there we have the first part of the derivative. Now for the second:
We can let
Substituting back in, we get:
Now we know the value of
To find the critical points, we set this expression equal to 0 and solve for x.
Now, this equation is a very hard one to solve, and I'm not the one to explain how to solve it. But I can tell you that the only real solution is around