17. Applications of Derivatives question bank (Maths)

If \( f : \mathbb{R} \to \mathbb{R} \), \( g : \mathbb{R} \to \mathbb{R} \) and \( f(x) + f''(x) = -xg(x)f'(x) \) and \( g(x) > 0 \,\, \forall\, x \in \mathbb{R} \) then \( f^2(x) + (f'(x))^2 \) has

Let \( h \) be a twice continuously differentiable positive function on an open interval \( J \). Let \( g(x) = \ln(h(x)) \) for each \( x \in J \). Suppose \( (h''(x))^2 > h'''(x)h(x) \) for each \( x \in J \). Then