​ Let ƒ : (a,b) → R be twice differentiable function such that ƒ(x) = ∫ g(t) dt x ∈[x,a] for a differentiable function g(x). ​

Question

Let ƒ : (a,b) → R be twice differentiable function such that ƒ(x) = \(\displaystyle\int _{a}^{x}\)g(t) dt for a differentiable function g(x). If ƒ(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least : 

(1) twelve roots in (a, b) 

(2) five roots in (a, b) 

(3) seven roots in (a, b) 

(4) three roots in (a, b)

Answer 1

f(x) = \(\displaystyle\int _{a}^{x}\)g(t)dt

f(x) → 5

f'(x) → 4

g(x) → 4

g'(x) → 3