Question

For the curve $x^2{y}^3=(2x+3y{)}^5,\frac{dy}{dx}=\frac{-y}{g(x)}$ where $g(x)$ is a real valued function. Define $h(x)=2g(x)+3(g(x){)}^{\frac{2}{3}}$.
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, $\frac{dy}{dx}=\frac{y}{x}$
$\Rightarrow g(x)=-x$
$\therefore h(x)=-2x+3x^{2/3}$
$h^{\prime }(x)=-2+\frac{2}{x^{1/3}}=2\left(\frac{1}{x^{1/3}}-1\right)$
As, $h^{\prime }\left(1^{+}\right)<0$ and $h^{\prime }\left(1^{-}\right)>0$, so $x=1$ is a point of maxima.