"Find the values of " x" that give relative extrema for: " f(x)=3x^5-5x^3

Answer 1

In general a 5th degree polynomial may have 4 turning points, each a local extremum. But this binomial has 2 terms and only 3 zeros. One zero is the origin (0,0) It has multiplicity 3, leaving just 2 local extrema, one positive, one negative.

the positive extremum will be the local minimum (1,-2), the bottom of a U shaped area of the curve

the negative extremum will be the local maximum (-1,2), the top of an inverted U shaped curve.

the origin (0,0) is an inflection point, neither local max nor min, it's where the 2nd derivative changes sign, where the inverted U shape changes to a U shape

f'(x) = 15x^4-15x^2 = 15x^2(x^2-1)= 0

set each factor = 0 and solve for x

x^2=0, x = 0

x^2-1 = 0,

x^2=1, x=+ or - sqr(1) = + or -1

Answer 2

f'(x) = 15x⁴ - 15x² = 0; x = 0; x = 1, x = - 1

f''(x) = 60x³ - 30x;

f''(0) = 0 so at x = 0 no local extrima;

f''(1) = 60 - 30 = 30 > 0 local min

f''(- 1) = -60 + 30 = 30 > 0 local max