In the carbonate ion (\( {CO}_3^{2-} \)), we can calculate the sum of the oxidation numbers as follows:
- The oxidation number of oxygen in the carbonate ion is \( -2 \), as oxygen generally has an oxidation state of \( -2 \) in compounds.
- Let the oxidation number of carbon be \( x \).
- Since the overall charge on the carbonate ion is \( -2 \), we can set up the following equation for the sum of oxidation numbers:
\[ x + 3(-2) = -2. \] Simplifying this equation: \[ x - 6 = -2 \quad \Rightarrow \quad x = +4. \] Thus, the oxidation number of carbon is \( +4 \), and the oxidation numbers of the three oxygen atoms are each \( -2 \).
Now, summing these oxidation numbers: \[ +4 + 3(-2) = +4 - 6 = -2. \]
Therefore, the sum of the oxidation numbers of all the atoms in the carbonate ion is \( -2 \).
Thus, the correct answer is \(-2\), which corresponds to option (C).
\[ f(x) = \begin{cases} x\left( \frac{\pi}{2} + x \right), & \text{if } x \geq 0 \\ x\left( \frac{\pi}{2} - x \right), & \text{if } x < 0 \end{cases} \]
Then \( f'(-4) \) is equal to:If \( f'(x) = 4x\cos^2(x) \sin\left(\frac{x}{4}\right) \), then \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:
Let \( f(x) = \frac{x^2 + 40}{7x} \), \( x \neq 0 \), \( x \in [4,5] \). The value of \( c \) in \( [4,5] \) at which \( f'(c) = -\frac{1}{7} \) is equal to:
The general solution of the differential equation \( \frac{dy}{dx} = xy - 2x - 2y + 4 \) is:
\[ \int \frac{4x \cos \left( \sqrt{4x^2 + 7} \right)}{\sqrt{4x^2 + 7}} \, dx \]