# What are the values and types of the critical points, if any, of #f(x) = (x^4/5)(x-4)^2#?

Turning points at

Flat, at x = Look at the two graphs.

It is easy to show that

Further, f'' is 0 only at x = 0. Yet, f''' = 0 here but

the next ( even order ) f'''' is not.

So, origin is not a point of inflexion, and so, at this turning point,

This flatness is zoomed in the second graph.

graph{5y-x^4(x-4)^2=0 [-40, 40, -20, 20]}

graph{5y-x^4(x-4)^2=0 [-5, 5, -2.5, 2.5]}

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To find the critical points of ( f(x) = \frac{x^{4/5}(x-4)^2}{5} ), we first need to find its derivative and then solve for ( x ) when the derivative is equal to zero.

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

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

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

The critical points occur when ( f'(x) = 0 ). Solving this equation may be complicated, but it can be simplified by factoring out common terms. After solving for ( x ), we can then classify the critical points as maxima, minima, or points of inflection by analyzing the second derivative or by examining the behavior of the function around these points.

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To find the critical points of ( f(x) = \frac{x^{4/5}(x-4)^2}{5} ), we first need to find its derivative and then solve for where the derivative is equal to zero.

( f'(x) = \frac{d}{dx}[\frac{x^{4/5}(x-4)^2}{5}] )

Using the product rule and the power rule, we get:

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

To find critical points, we set the derivative equal to zero:

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

After solving this equation, we can find the values of ( x ) corresponding to the critical points. Then we can determine the types of critical points by analyzing the behavior of the function around those points.

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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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