If \(a, b\) be positive and \(a+b=1(a \neq b)\) and If \(A=\sqrt[3]{a}+\sqrt[3]{b}\) then the correct statement is (1) \(A>2^{2 / 3}\) (2) \(\mathrm{A}=\frac{2^{2 / 3}}{3}\) (3) \(\mathrm{A}<2^{2 / 3}\) (4) \(A=2^{2 / 3}\)

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a fundamental concept in algebra. It states that for any non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. Mathematically, for two numbers, it can be written as: \ \[ \frac{a + b}{2} \geq \sqrt{ab} \] In our problem, we use this to compare the sum and product of two positive numbers. Given that \(a + b = 1\), applying AM-GM gives: \ \[ \frac{1}{2} \geq \sqrt{ab} \] By squaring both sides, we infer: \ \[ \frac{1}{4} \geq ab \] The AM-GM inequality is a powerful tool in solving various algebraic inequalities by providing a pathway to compare sums and products in a useful manner.

Cube roots are essential to this problem. The cube root of a number \(x\) is a value that when multiplied by itself three times gives \(x\). Mathematically, the cube root of \(x\) is represented as \( \sqrt[3]{x} \). In the problem, we use cube roots of both \(a\) and \(b\) to find \(A\). Applying the AM-GM inequality to cube roots, we get: \ \[ \frac{ \sqrt[3]{a} + \sqrt[3]{b} }{2} \geq \sqrt[3]{ \sqrt[3]{ab} } \] To evaluate the right side, we consider \( \sqrt[4]{ab} \) and use previous results (\(ab \leq \frac{1}{4}\)): \ \[ \left( \frac{1}{4} \right)^{\frac{1}{3}} = 2^{ -\frac{2}{3} } \] Thus, \ \[ \sqrt[3]{a} + \sqrt[3]{b} \geq 2 \cdot 2^{ -\frac{2}{3} } \ (using the AM-GM inequality) \] This helps us to bound \(A\).

Algebraic inequalities involve expressions with inequality signs (\(>\), \(<\), \( \geq \), \( \leq \)). In this exercise, we combine various inequalities to get the answer. Beginning with \( a + b = 1 \) and \( a eq b \), we used AM-GM inequality: \ \[ \frac{1}{2} \geq \sqrt{ab} \] leading to: \ \[ \frac{1}{4} \geq ab \] Further applying inequality on cube roots: \ \[ \frac{ \sqrt[3]{a} + \sqrt[3]{b} }{2} \geq \sqrt[3]{ \sqrt[3]{ab} } \] By solving it, we derived: \ \[ \sqrt[3]{a} + \sqrt[3]{b} \geq 2 \cdot 2^{ -\frac{2}{3} } \rightarrow 2^{ \frac{2}{3} } \] The final strict inequality, owing to \( a eq b \), indicates: \ \[ \mathrm{A} < 2^{ \frac{2}{3} } \] This output confirms option (3) as correct. Inequalities help break down complex problems into simpler comparisons.