Statement-1 : For 0` le p lt 1` and for any positive a and b the intequality `(a+b)^p lt a^p+b^p` is valid
Staement - 2: `For 0 le ple 1` the function `f(x)=1 +x^p -(1+x)^p` decreases on `[0,oo)`

Options

A. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is a correct explanation for Statement-7
B. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is not a correct explanation for Statement-7
C. Statement-1 True statement -1 is True,Statement -2 is False
D. Statement-1 is False ,Statement -2 is True

Explanation

To solve the problem, we need to analyze both statements provided in the question.

### Step 1: Analyze Statement 1
**Statement 1:** For \(0 \leq p < 1\) and for any positive \(a\) and \(b\), the inequality \((a+b)^p < a^p + b^p\) is valid.

To verify this statement, we can use the properties of convex functions. The function \(f(x) = x^p\) for \(0 < p < 1\) is a concave function. By Jensen's inequality, for a concave function, we have:

\[ f\left(\frac{a+b}{2}\right) \geq \frac{f(a) + f(b)}{2} \]

This can be rearranged to show that:

\[ (a+b)^p < a^p + b^p \]

Thus, Statement 1 is **true**.

### Step 2: Analyze Statement 2
**Statement 2:** For \(0 \leq p \leq 1\), the function \(f(x) = 1 + x^p - (1+x)^p\) decreases on \([0, \infty)\).

To analyze this, we need to find the derivative of \(f(x)\):

\[ f'(x) = \frac{d}{dx}[1 + x^p - (1+x)^p] \]

Using the power rule and chain rule, we get:

\[ f'(x) = p x^{p-1} - p(1+x)^{p-1} \]

Factoring out \(p\):

\[ f'(x) = p \left( x^{p-1} - (1+x)^{p-1} \right) \]

Now, we need to determine the sign of \(f'(x)\) over the interval \([0, \infty)\).

1. For \(x = 0\):
\[ f'(0) = p(0^{p-1} - 1^{p-1}) = p(0 - 1) = -p < 0 \quad (\text{since } p > 0) \]

2. For \(x > 0\):
Since \(0 \leq p \leq 1\), \(x^{p-1}\) is less than or equal to \((1+x)^{p-1}\) for all \(x > 0\). Thus, \(f'(x) < 0\).

Since \(f'(x) < 0\) for all \(x \geq 0\), the function \(f(x)\) is decreasing on \([0, \infty)\).

Thus, Statement 2 is **true**.

### Conclusion
- Statement 1 is **true**.
- Statement 2 is **true**.

### Final Answer
Both statements are true.