Question

For the curve $x^2 y^3=(2 x+3 y)^5, \frac{d y}{d x}=\frac{-y}{g(x)}$ where $g(x)$ is a real valued function. Define $h(x)=2 g(x)+3(g(x))^{\frac{2}{3}}$.
Which one of the following statement is correct for the function $h ( x )$ ?

• A. $x =-1$ is the point of maxima.
• B. $x=1$ is the point of maxima.
• C. Non-derivable at $x =-1$.
• D. $x=0$ is point of maxima.

Answer 1

Answer: (B)

As given equation is homogeneous so, $\frac{d y}{d x}=\frac{y}{x}$
$\Rightarrow g ( x )=- x $
$\therefore h ( x )=-2 x +3 x ^{2 / 3}$
$h ^{\prime}( x )=-2+\frac{2}{ x ^{1 / 3}}=2\left(\frac{1}{ x ^{1 / 3}}-1\right)$
As, $h ^{\prime}\left(1^{+}\right)<0$ and $h ^{\prime}\left(1^{-}\right)>0$, so $x =1$ is a point of maxima.