How do you find all critical numbers for the function #f(x)= (x + 2)^3*(x -1)^4#?

Answer 1

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:

#x^y=z#

Then we can use a logarithm to "cancel out" the exponent, bringing it to the front like so:

#ylnx=lnz#
Now, we don't have to use the natural logarithm (#lnx#) all the time - we could use, say, #log_2(x)# - but mathematicians think the natural log is cool (and it's extremely useful when differentiating, as we'll see in a moment).
Okay, so what's the point? Well, according to the rules of logs, if we take the log of both sides: #ln(f(x))=ln((x+2)^3(x-1)^4)#
We can split the multiplication into addition: #ln(f(x))=ln(x+2)^3+ln(x-1)^4#
And remember, exponents come down to the front: #ln(f(x))=3ln(x+2)+4ln(x-1)#
Now we can take the derivative, making sure to use the chain rule on the left side: #(f'(x))/f(x)=3/(x+2)+4/(x-1)-># because the derivative of #lnx# is #1/x#
Now just multiply both sides by #f(x)#: #f'(x)=(3/(x+2)+4/(x-1))f(x)#
And since #f(x)=(x+2)^3(x-1)^4#: #f'(x)=(3/(x+2)+4/(x-1))((x+2)^3(x-1)^4)#
Finally, set #f'(x)=0# and solve: #0=(3/(x+2)+4/(x-1))((x+2)^3(x-1)^4)#
We have three solutions: #0=3/(x+2)+4/(x-1)# and #0=(x+2)^3# and #0=(x-1)^4#
The last two clearly mean that #x=-2# and #x=1# are solutions. For the first, we set up the equation: #-3/(x+2)=4/(x-1)# #->-3(x-1)=4(x+2)#
Which is a simple equation that yields #x=-5/7#.

So there you have it: same answer, different method. Use which one you feel most comfortable with - that's the beauty of math.

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Answer 2

To find all critical numbers for the function ( f(x) = (x + 2)^3 \cdot (x - 1)^4 ), you need to first find its derivative, ( f'(x) ). Then, solve for ( x ) where ( f'(x) = 0 ) and where ( f'(x) ) is undefined. These points will be the critical numbers of the function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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