Differentiation and its Application: The function, f(x)=(3x-7)x^2/3 , xinmathbfR , is increasing for all… | JEE Main 2020

JEE Main 2020 (Online) 3rd September Morning ShiftQuestion 1

The function, $f(x)=(3x-7)x^{2/3}$, $x\in\mathbf{R}$, is increasing for all $x$ lying in :

Differentiation and its Application

$\left(-\infty,\frac{14}{15}\right)$$\left(-\infty,-\frac{14}{15}\right)\cup(0,\infty)$$(-\infty,0)\cup\left(\frac{14}{15},\infty\right)$✓ Correct answer$(-\infty,0)\cup\left(\frac37,\infty\right)$

Solution

Concept. Write $f'$ as one fraction; numerator and denominator change sign at $\frac{14}{15}$ and $0$. Formulas used. (1) $f=3x^{5/3}-7x^{2/3}$ $\Rightarrow$ $f'=5x^{2/3}-\dfrac{14}{3}x^{-1/3}=\dfrac{15x-14}{3x^{1/3}}$Step 1 - Sign. By (1), $f'>0$ when $15x-14$ and $x^{1/3}$ have the same sign: $x<0$ or $x>\frac{14}{15}$. Answer: (C)

Differentiation and its Application - JEE Mathematics question (JEE Main 2020 (Online) 3rd September Morning Shift)

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