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Step 1: Understand the Given Information
We have a function $f(x) = \int_a^x g(t)\,dt$ on the interval $(a, b)$, where $g(x)$ is differentiable. It is also given that $f(x) = 0$ has exactly five distinct roots in $(a, b)$. We need to find the minimum number of roots of the equation $g(x)\,g'(x) = 0$ in $(a, b)$.
Step 2: Express the Derivatives
Since $f(x) = \int_a^x g(t)\,dt$, by the Fundamental Theorem of Calculus, we get:
\[
f'(x) = g(x),
\quad
f''(x) = g'(x).
\]
Therefore, the given condition $g(x)\,g'(x) = 0$ can also be written as:
\[
f'(x)\,f''(x) = 0.
\]
This implies that for each solution, either $f'(x) = 0$ or $f''(x) = 0$ at that point.
Step 3: Apply Rolle’s Theorem to Count the Roots
We know $f(x) = 0$ has exactly five distinct roots in $(a, b)$. Denote these roots by
\[
x_1 < x_2 < x_3 < x_4 < x_5.
\]
By Rolle’s Theorem, between any two consecutive roots of $f(x)$, there is at least one root of $f'(x)$. Hence, if $f(x)$ has 5 distinct roots, $f'(x)$ must have at least 4 distinct roots:
\[
f'(x) \text{ has at least 4 distinct roots.}
\]
Step 4: Further Application to Count Roots of }f''(x)
Similarly, applying Rolle’s Theorem to $f'(x)$, which has at least 4 distinct roots, we find that $f''(x)$ must have at least 3 distinct roots:
\[
f''(x) \text{ has at least 3 distinct roots.}
\]
Step 5: Combine the Results for }f'(x)\,f''(x) = 0
The equation $f'(x)\,f''(x) = 0$ is satisfied if either $f'(x) = 0$ or $f''(x) = 0$. Since $f'(x)$ has at least 4 distinct roots and $f''(x)$ has at least 3 distinct roots, the product $f'(x)f''(x) = 0$ must have at least
\[
4 + 3 = 7
\]
distinct roots (counting all the distinct points where either $f'(x)$ or $f''(x)$ becomes zero).
Step 6: Relate Back to }g(x)\,g'(x) = 0
Recalling that $g(x) = f'(x)$ and $g'(x) = f''(x)$, we see that:
\[
g(x)\,g'(x) = f'(x)\,f''(x).
\]
Therefore, $g(x)\,g'(x) = 0$ also has at least 7 roots in $(a, b)$.
Final Answer
The minimum number of roots of $g(x)\,g'(x) = 0$ in $(a,b)$ is 7.
