How do you find all critical numbers for the function #f(x)= (x + 2)^3*(x -1)^4#?
Alternatively, we can get the same result by using logarithmic differentiation.
Remember the basic concept of the logarithm: it is the opposite of exponentiation. That is to say, if:
Then we can use a logarithm to "cancel out" the exponent, bringing it to the front like so:
Now, we don't have to use the natural logarithm (
Okay, so what's the point? Well, according to the rules of logs, if we take the log of both sides:
We can split the multiplication into addition:
And remember, exponents come down to the front:
Now we can take the derivative, making sure to use the chain rule on the left side:
Now just multiply both sides by
We have three solutions:
The last two clearly mean that
Which is a simple equation that yields
So there you have it: same answer, different method. Use which one you feel most comfortable with - that's the beauty of math.