Determine the intervals where fx 10x4x5 is increasing and d

Determine the intervals where f(x) = 10x4-x5 is increasing and decreasing, and identify any local extrema.

Solution

3)
f(x) = sin^4(5x^2+2)
f\'(x) = 4sin^3(5x^2+2)*cos(5x^2+2)*10x
f\'(x) = 40xcos(5x^2+2)sin^3(5x^2 + 2)
4)
f(x) = 10x^4 - x^5
f\'(x) = 40x^3 - 5x^4
for increasing f\'(x) > 0
40x^3 - 5x^4 > 0
5x^3(8-x) > 0
x(x-8) < 0
0 < x < 8........increasing interval
x<0, x>8.....decreasing interval
critical points are x = 0, 8
f\'\'(x) = 120x^2 - 20x^3
f\'\'(0) = 0
f\'\'(8) = -ve
hence x = 8 has local maxima & x = 0 is the inflection point