Where does the graph of y = ( 5 x 4 ) - ( x 5 ) have an inflection point?

Question

Where does the graph of       y    =          (      5              x        4            )        -          (              x        5            )       have an inflection point?

Dakota George · Accepted Answer

An inflection point on a graph is a point where the concavity changes. We'll look at the sign of the second derivative to investigate concavity:

y = 5 x^{4} - x^{5}

y' = 20 x^{3} - 5 x^{4}

y'' = 60 x^{2} - 20 x^{3} = 20 x^{2} (3 - x)

Obviously

20 x^{2}

is always positive, so the sign of y'' is the same as the sign of

3 - x

.
Which is positive for

x < 3

and negative for

x > 3

. At

x = 3

the concavity changes.
Because an inflection point is a point on a graph, we need:
when

x = 3

, we get

y = 5 (3^{4}) - 3^{5} = 5 (3^{4}) - 3 (3^{4}) = 2 (3^{4}) = 2 (81) = 162

The point (3, 162) in the only inflection point for the graph of

y = 5 x^{4} - x^{5}

Where does the graph of y=(5x^4)-(x^5) have an inflection point?

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