critical numbers

Question

y= 5x^2/3 + x^5/3

Answer 1

Critical numbers are where the 1st derivative equals zero or is undefined.

y' = (10/3)x^-1/3 + (5/3)x^2/3 (note that this is undefined when x = 0--division by zero)

10/3x^1/3 + 5x^2/3/3 = 0 (now multiply every term by 3x^1/3)

10 + 5x = 0

x = -2

So the critical numbers are 0 and -2

This graph shows that there is a horizontal tangent line at the point where x = -2 and a cusp at the point where x = 0.

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