Select the graph that schematically represents BOTH y = x^^m and y = x^^1/m properly in the interval 0 ≤ x ≤ 1. For integer values of m, where m > 1

Correct Answer/Option:
The correct option is the first option, which corresponds to the image dj6466-1721300611764.jpg.

Detailed Step-by-Step Explanation:

We are asked to identify the graph that schematically represents both curves:
$y=x^m$
and
$y=x^{1/m}$
properly in the interval:
$0\le x\le 1$
for integer values of $m$ where $m>1$.

Let's analyze the properties of these two functions within the given interval step-by-step:

1. Coordinates of the Endpoints
At the lower boundary $x=0$:
$y=0^m=0$
and
$y=0^{1/m}=0$
Thus, both curves start at the origin $(0,0)$.

At the upper boundary $x=1$:
$y=1^m=1$
and
$y=1^{1/m}=1$
Thus, both curves terminate at the point $(1,1)$.

2. Comparing Values in the Open Interval $(0,1)$
Since $m>1$ is an integer, for any value of $x$ strictly between $0$ and $1$:
- Multiplying $x$ by itself $m$ times reduces the value, so $x^m<x$.
- Taking the $m$-th root of $x$ increases the value, so $x^{1/m}>x$.
This gives the relation:
$x^m<x<x^{1/m}$
Therefore, the graph of the curve $y=x^{1/m}$ must lie entirely above the line of identity $y=x$, while the graph of the curve $y=x^m$ must lie entirely below it.

3. Slope and Curvature at the Origin $(0,0)$
Let us evaluate the derivatives to determine the slopes at the origin:
- For the lower curve:
$\frac{dy}{dx}=\frac{d}{dx}(x^m)=m{x}^{m-1}$
Since $m>1$, at $x=0$, the derivative is:
${(\frac{dy}{dx}}_{x=0}=0$
This means $y=x^m$ starts with a flat horizontal tangent at the origin and curves upward (concave up).

- For the upper curve:
$\frac{dy}{dx}=\frac{d}{dx}(x^{1/m})=\frac{1}{m}{x}^{\frac{1}{m}-1}=\frac{1}{m{x}^{1-1/m}}$
Since $1-1/m>0$, as $x\to 0^{+}$, the denominator approaches 0, meaning:
$\underset{x\to 0^{+}}{lim}\frac{dy}{dx}=\mathrm{\infty }$
This means $y=x^{1/m}$ starts with a vertical tangent at the origin and curves downward (concave down).

4. Symmetry and Inverse Functions
The two functions are mathematical inverses of each other in the domain $[0,1]$. Thus, their graphs are perfectly symmetric reflections of each other across the diagonal line $y=x$.

Conclusion:
In the correct graph (Option 1, corresponding to the image dj6466-1721300611764.jpg):
- The horizontal axis is $x$ going from $0$ to $1$, and the vertical axis is $y$ going from $0$ to $1$.
- The upper curve (rising vertically at the start, concave down) is correctly labeled as $y=x^{1/m}$.
- The lower curve (starting horizontally flat, concave up) is correctly labeled as $y=x^m$.