Calculus problem?

Question

Let f be the real-valued function defined by f(x) = sin^3x + sin^3 |x|.
a) Find f'(x) for x > 0.
b) Find f'(x) for x < 0.
c) Determine whether f(x) is continuous at x = 0. Justify your answer.
d) Determine whether the derivative of f(x) exists at x = 0. Justify your answer.

Answer 1

f(x) = 2sin^3(x) for x in [0,pi]

f(x) = 0 for x in [-pi,0]

Proceed.

Answer 2

f (x) = x^3 - 12x + a million . . . the 1st spinoff set to 0 unearths turning or table sure factors f ' (x) = 3x^2 - 12 3x^2 - 12 = 0 3 * (x + 2) * (x - 2) = 0 x = 2 ... x = - 2 . . . the 2nd spinoff evaluated at x = 2 and -2 determines if those factors are min, max, or neither. f ' ' (x) = 6x f ' ' (2) = 6*2 = 12 <== effective fee shows x=2 is a close-by minimum f ' ' (-2) = 6*(-2) = -12 <== unfavorable fee shows x=-2 is a close-by maximum a.) x = - 2 is a maximum, and x=2 is a minimum ... so x = - infinity to -2 is increasing x = -2 to +2 is lowering x = +2 to + infinity is increasing b.) f (-2) = (-2)^3 - 12*(-2) + a million = 17 f (2) = (2)^3 - 12*(2) + a million = - 15 c.) . . . the 2nd spinoff set to 0 unearths inflection factors, or the place concavity differences 6x = 0 x = 0 <=== inflection element x = - 2 is a maximum, so could be concave down concavity differences on the inflection element(s) ... so x = - infinity to 0 is concave down x = 0 to + infinity is concave up