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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
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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