NIMCET 2024 Paper

Section: Mathematics

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Question : 34 of 120

Marks: +1, -0

Consider the function

f(x)=x^{2 \/ 3}(6-x)^{1 \/ 3}

. Which of the following statements is false?

$f$ is increasing in interval $(0,4)$
$f$ is decreasing in interval $(6, ∞)$
$f$ has a point of inflection at $x=0$
$f$ has a point of inflection at $x=6$

Solution:

\;f(x)=x^{2 \/ 3}(6-x)^{1 \/ 3}

\;f^{'}(x)=\;{2}/{3} x^{-\;{1}/{3}}(6-x)^{\;{1}/{3}}-\;{1}/{3}(6-x)^{-\;{2}/{3}} x^{\;{2}/{3}}

\;=(6-x)^{-\;{2}/{3}} x^{-\;{1}/{3}}(4-x)

We see that the value of

f^{'}(x)>0

in the interval

(0,4)

, also the value of

f^{'}(x)< 0

when

x>6

.
Also

f^{' '}(x)=0

at

x=6

, hence

f(x)

has a point of inflection at

x=6

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