Raymond B. answered • 03/08/21
Math, microeconomics or criminal justice
Potentially one less than the degree
with degree 6, the maximum potential number of turning points is 6-1 = 5
but with just two terms, there's less.
take the derivative, set it equal to zero. That gives the turning points, where slope =0
f'(x) = 5x^4 + 12x^5 = 0
x^4(5+12x) =0
x=0 and x = -5/12 are the x values of the turning points, just 2 turning points, at the origin and at
(-5/12, -3125/1492992) or (-5/12, -0.00293112354) those 2 turning points are the local extrema
(-5/12)^5 + 2(-5/12)^6 = -0,.00293112354,
but take the 2nd derivative f''(x) = 20x^3 + 60x^4 f''(0)=0, the origin isn't a turning point, as f(x) continues to increase before and after. Only for an instant does the graph not go up.
the other point (-5/12, -3125/1492992) is a local and global minimum and the only real turning point.
But had there been more terms there could have potentially been 5 turning points.
