f[x_] = x^4 Sin[x + 1] - x^2

f'[x]

Plot[f[x], {x, -2, 2}]

Plot[f'[x], {x, -2, 2}]

{f'[-1], f'[0], f'[0.5], f'[1], f'[2]} // 
      N

x1 = FindRoot[f'[x] == 0, {x, 0, -0.5, 0.5}]

x2 = FindRoot[f'[x] == 0, {x, 0.5, 0.1, 1}]

x3 = FindRoot[f'[x] == 0, {x, 1.5, 1, 2}]

{x1, x2, x3}

Print["Monotonically increasing interval is: ", {-2, x1}, {x2, 
        x3}]
Print["Monotonically decreasing interval is: ", {x1, x2}, {x3, 
        2}]