Let $f: (a,b) \to \mathbb{R}$ be twice differentiable function such that $f(x) = \int_a^x g(t)dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g'(x)=0$ has at least :
Show Hint
Rolle's Theorem is a powerful tool for finding the number of roots of derivatives. If a function $h(x)$ has $n$ roots, its derivative $h'(x)$ must have at least $n-1$ roots located between the roots of $h(x)$.
Given:
\[
f(x) = \int_a^x g(t)\,dt
\Rightarrow f'(x)=g(x), \quad f''(x)=g'(x)
\]
It is given that \( f(x)=0 \) has exactly five distinct roots in \( (a,b) \).
Let them be:
\[
a < c_1 < c_2 < c_3 < c_4 < c_5 < b
\]
Step 1: Zeros of \( g(x) \)
Between each pair of consecutive roots of \( f(x) \),
Rolle’s theorem guarantees at least one root of \( f'(x)=g(x) \).
Hence, \( g(x)=0 \) has at least:
\[
5 - 1 = 4 \text{ distinct roots}
\]
Step 2: Zeros of \( g'(x) \)
Between consecutive roots of \( g(x) \),
Rolle’s theorem again guarantees roots of \( g'(x) \).
Thus, \( g'(x)=0 \) has at least:
\[
4 - 1 = 3 \text{ distinct roots}
\]
Step 3: Zeros of \( g(x)g'(x)=0 \)
The equation \( g(x)g'(x)=0 \) is satisfied when either:
\[
g(x)=0 \quad \text{or} \quad g'(x)=0
\]
Total minimum number of distinct roots:
\[
4 + 3 = 7
\]
\[
\boxed{\text{At least } 7 \text{ roots}}
\]
