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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R,$ is increasing for all $x$ lying in
  • A
    $(-\infty, 0) \cup\left(\frac{3}{7}, \infty\right)$
  • ✓
    $(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$
  • C
    $\left(-\infty, \frac{14}{15}\right)$
  • D
    $\left(-\infty,-\frac{14}{15}\right) \cup(0, \infty)$
✓

Answer

Correct option: B.
$(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$
b
$f(x)=(3 x-7) x^{2 / 3}$

$\Rightarrow \quad f(x)=3 x^{5 / 3}-7 x^{2 / 3}$

$\Rightarrow \quad f^{\prime}(x)=5 x^{2 / 3}-\frac{14}{3 x^{1 / 3}}$

$=\frac{15 x-14}{3 x^{1 / 3}}>0$

$\therefore \quad f ^{\prime}( x )>0 \forall x \in(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$

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