Let f:(a, b) R be twice differentiable function such that f(x)= _…

MCQ

Let $f:(a, b) \rightarrow R$ be twice differentiable function such that $f(x)=\int_{a}^{x} g(t) d t$ for a differentiable function $g(x) .$ If $f(x)=0$ has exactly five distinct roots in $(a, b)$, then $g(x) g^{\prime}(x)=0$ has at least:

✓
seven roots in $(a, b)$
B
five roots in $(a, b)$
C
three roots in $(a, b)$
D
twelve roots in $(a, b)$

✓

Answer

Correct option: A.

seven roots in $(a, b)$

a
$\mathrm{f}(\mathrm{x})=\int_{\mathrm{a}}^{\mathrm{x}} \mathrm{g}(\mathrm{t}) \,\mathrm{d} \mathrm{t}$

$\mathrm{f}(\mathrm{x}) \rightarrow 5$

$\mathrm{f}^{\prime}(\mathrm{x}) \rightarrow 4$

$\mathrm{~g}(\mathrm{x}) \rightarrow 4$

$\mathrm{~g}^{\prime}(\mathrm{x}) \rightarrow 3$

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