Let f:[0,1]—> R be a thrice differentiable function such that f(0)= f (1) and f'(½)=f'(¾). Van f'''(x⁰) =0 for some x⁰=(0'1)

f: [-5,5]→[3,18] Surjective & differentiable f

g(x) = f(x) f'(x) Twice differentiable fr

g(0)=k Find least integral value of k such that there always exists some CEES✓ f" to satisfy (f' * (c)) ^ 2 =-f(c)f^ prime prime (c) \&(f(c))^ 2 f^ prime prime (c) < 3 * (f' * (c)) ^ 3