What are the critical values, if any, of #f(x)=x^(4/5) (x − 3)^2#?

Answer 1

They are #0#, #3#, and #6/7#.

#f(x)=x^(4/5) (x − 3)^2#
Critical value of #f# are values in the domain of #f# at which either #f'# does not exist or #f'(x)=0#
The domain of #f# is #RR#.
So every point at which #f'# fails to exist or #f'(x)=0# is a critical value for #f#.

Differentiate using the product rule:

#f'(x) = 4/5 x^(-1/5)(x-3)^2+x^(4/5) 2(x-3)(1)# #" "# (using the chain rule at the end)
#f'(x) = (4(x-3)^2)/(5root(5)x)+(2root(5)x^4(x-3))/1#
# = (4(x-3)^2+10x(x-3))/(5root(5)x)#
# = (2(x-3)[2(x-3)+5x])/(5root(5)x)#
# = (2(x-3)(7x-6))/(5root(5)x)#
#f'(0)# does not exist, but #0# is in the domain of #f#, so #0# is a critical value for #f#.
#f'(x) = 0# at #x=3# and at #x=6/7#, both of which are in the domain of #f#.
The critical values are #0#, #3#, and #6/7#.
We have finished the problem without looking at the graph of #f#. Having done that, it can be helpful to look at the graph. You can zoom in and out and drag the graph around using a mouse. (It will start the same every time you return to this answer.)

graph{y=x^(4/5)(x-3)^2 [-3.01, 6.857, -0.442, 4.49]}

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Answer 2

To find the critical values of ( f(x) = x^{\frac{4}{5}} (x - 3)^2 ), we need to first find the derivative of ( f(x) ), set it equal to zero, and solve for ( x ).

Taking the derivative of ( f(x) ) using the product rule:

( f'(x) = \frac{4}{5}x^{-\frac{1}{5}}(x-3)^2 + x^{\frac{4}{5}}(2(x-3)) )

Setting ( f'(x) ) equal to zero and solving for ( x ):

( \frac{4}{5}x^{-\frac{1}{5}}(x-3)^2 + x^{\frac{4}{5}}(2(x-3)) = 0 )

( \frac{4}{5}x^{-\frac{1}{5}}(x-3)^2 = -2x^{\frac{4}{5}}(x-3) )

( \frac{4}{5}(x-3) = -2x^{\frac{4}{5}} )

( 4(x-3) = -\frac{10}{5}x^{\frac{4}{5}} )

( 4x - 12 = -2x^{\frac{4}{5}} )

( 2x^{\frac{4}{5}} + 4x - 12 = 0 )

This equation can be solved numerically to find the critical values.

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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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