Question

The number of real solution(s) of the equation $x^{1/3}+\sqrt{x}=1$ is

• A. $2$
• B. $3$
• C. $1$
• D. No real solution

Answer 1

The correct option is C $1$

We know that $x$ cannot be negative .
Let $y_1=1-\sqrt{x}$ and $y_2=x^{1/3}$
We need to find the intersection between this two curves.
$y_1^{\prime }=-\frac{1}{2\sqrt{x}}<0$ so $y_1$ is monotonic decreasing function.
$y_2^{\prime }=\frac{1}{3}{x}^{-2/3}>0$ so $y_2$ is a monotonic increasing function.
Plotting this two curves,

Hence, only one solution of the given equation is possible.