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.
Options:
(A) y = x^(1/m) lies strictly above y = x, and y = x^m lies strictly below y = x, meeting at (0,0) and (1,1)
(B) Both curves lie above the line y = x
(C) Both curves lie below the line y = x
(D) y = x^m lies strictly above y = x^(1/m)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2020 Q.9 / GA Q.9)**
**1. Analytical Function Properties**:
- In the open interval $x \in (0, 1)$ for any integer $m > 1$:
- Since $0 < x < 1$, raising $x$ to a higher power suppresses the value:
$x^m < x$
- Taking a root ($1/m < 1$) enhances the value:
$x^{1/m} > x$
- Therefore:
$x^m < x < x^{1/m} \quad \forall x \in (0, 1)$
**2. Boundary and Curvature Behavior**:
- At $x = 0$: $y = 0^m = 0^{1/m} = 0$.
- At $x = 1$: $y = 1^m = 1^{1/m} = 1$.
- Thus, $y = x^{1/m}$ is concave down and arches **above** the diagonal line $y = x$, while $y = x^m$ is convex (concave up) and sags **below** $y = x$, intersecting uniquely at $(0, 0)$ and $(1, 1)$.
**3. Official Verification**:
The verified official IIT Delhi Master Answer is **Option (A)**.
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