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

Question

Cameron Alexander · Accepted Answer

Turning points at #x = 8/3 and x = 4#.
Flat, at x = Look at the two graphs.

It is easy to show that #f' = 0, x = 0, 8/3 and 4#. 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, the curve is flat, in the #epsilon-#neighborhood #(-epsilon, epsilon)# 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]}

Aurora Alvarado · Answer

undefined 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.

Luke Alvarez · Answer

undefined 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.