Statement I The equation `(x^(3))/(4) - sin pi x + (2)/(3) = 0` has atleast one solution in [-2, 2].
Statement II Let `f : [a, b] rarr R` be a function and c be a number such that `f(a) lt c lt f(b)`, then there is atleast one number `n in (a, b)` such that f(n) = c.

Options

A. Statement I is correct, Statement II is also correct, Statement II is the correct explanation of Statement I
B. Statement I is correct, Statement II is also correct, Statement II is not the correct explanation of Statement I
C. Statement I is correct, Statement II is incorrect
D. Statement I is incorrect, Statement II is correct.

Answer

To solve the problem, we need to analyze both statements and determine their validity.

### Step 1: Define the function
Let us define the function based on the equation given in Statement I:
\[ f(x) = \frac{x^3}{4} - \sin(\pi x) + \frac{2}{3} \]
We need to check if this function has at least one root in the interval \([-2, 2]\).

### Step 2: Evaluate the function at the endpoints
Next, we will evaluate \(f(x)\) at the endpoints of the interval:
1. Calculate \(f(-2)\):
\[ f(-2) = \frac{(-2)^3}{4} - \sin(-2\pi) + \frac{2}{3} \]
\[ = \frac{-8}{4} - 0 + \frac{2}{3} = -2 + \frac{2}{3} = -\frac{6}{3} + \frac{2}{3} = -\frac{4}{3} \]

2. Calculate \(f(2)\):
\[ f(2) = \frac{(2)^3}{4} - \sin(2\pi) + \frac{2}{3} \]
\[ = \frac{8}{4} - 0 + \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \]

### Step 3: Apply the Intermediate Value Theorem
Now we have:
- \(f(-2) = -\frac{4}{3} < 0\)
- \(f(2) = \frac{8}{3} > 0\)

Since \(f(-2) < 0\) and \(f(2) > 0\), by the Intermediate Value Theorem, there exists at least one \(n \in (-2, 2)\) such that \(f(n) = 0\). Thus, Statement I is true.

### Step 4: Validate Statement II
Statement II states that if \(f(a) < c < f(b)\), then there exists at least one \(n \in (a, b)\) such that \(f(n) = c\). This is a direct consequence of the Intermediate Value Theorem, which is indeed true for continuous functions.

### Conclusion
Both statements are true:
- Statement I is true because the function has at least one root in \([-2, 2]\).
- Statement II is true as it describes the Intermediate Value Theorem.

Thus, the correct conclusion is that both statements are true, and Statement II provides a correct explanation for Statement I.

### Final Answer
Both Statement I and Statement II are true.

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