Let f(x) = _0^ x^2 ( t - 1 ) ( t - 4 ) ( t - 9 )dt , then - Vidyadip

MCQ

Let $f(x) = \int\limits_0^{{x^2}} {\left( {t - 1} \right)} \left( {t - 4} \right)\left( {t - 9} \right)dt$ , then

A
$f''(x) = 0$ have $4$ distinct positive solutions
✓
$f'''(x) = 0$ have $2$ distinct positive solutions
C
$f'''(x) = 0$ have $3$ distinct positive solutions
D
$f(x)$ have $6$ critical points.

✓

Answer

Correct option: B.

$f'''(x) = 0$ have $2$ distinct positive solutions

b
$f^{\prime}=2 x\left(x^{2}-1\right)\left(x^{2}-4\right)\left(x^{2}-9\right)$

$f(\mathrm{x})$ have $7$ critical points and $f^{\prime \prime}=0$ have $3$

positive and $3$ negative solutions $f^{\prime}$ is odd function there for $f^{\prime \prime \prime}$ is also and odd function so $f^{\prime \prime \prime}(0)=0$

as there are $5$ roots of $f^{\prime \prime \prime}(\mathrm{x})=0$ there for it will have $2$ positive and $2$ negative roots.

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