The number of real solutions of the equation `x^3+1=2\ root(3)(2x-1),` is

Explanation

To find the number of real solutions for the equation \( x^3 + 1 = 2 \sqrt[3]{2x - 1} \), we can follow these steps:

### Step 1: Define the Functions
Let us define two functions based on the given equation:
- \( f_1(x) = x^3 + 1 \)
- \( f_2(x) = 2 \sqrt[3]{2x - 1} \)

### Step 2: Analyze the Function \( f_1(x) \)
The function \( f_1(x) = x^3 + 1 \) is a cubic function. It is continuous and differentiable everywhere.

- For \( x = 0 \):
\[ f_1(0) = 0^3 + 1 = 1 \]
- For \( x = 1 \):
\[ f_1(1) = 1^3 + 1 = 2 \]
- For \( x = -1 \):
\[ f_1(-1) = (-1)^3 + 1 = 0 \]
- For \( x = -2 \):
\[ f_1(-2) = (-2)^3 + 1 = -7 \]

### Step 3: Analyze the Function \( f_2(x) \)
The function \( f_2(x) = 2 \sqrt[3]{2x - 1} \) is also continuous and differentiable.

- For \( x = 0 \):
\[ f_2(0) = 2 \sqrt[3]{2(0) - 1} = 2 \sqrt[3]{-1} = -2 \]
- For \( x = 1 \):
\[ f_2(1) = 2 \sqrt[3]{2(1) - 1} = 2 \sqrt[3]{1} = 2 \]
- For \( x = 2 \):
\[ f_2(2) = 2 \sqrt[3]{2(2) - 1} = 2 \sqrt[3]{3} \approx 3.17 \]
- For \( x = -2 \):
\[ f_2(-2) = 2 \sqrt[3]{2(-2) - 1} = 2 \sqrt[3]{-5} \approx -4.64 \]

### Step 4: Graph the Functions
Now, we can graph both functions \( f_1(x) \) and \( f_2(x) \):
- \( f_1(x) \) starts from \( -7 \) at \( x = -2 \), crosses \( 0 \) at \( x = -1 \), reaches \( 1 \) at \( x = 0 \), and goes up to \( 2 \) at \( x = 1 \).
- \( f_2(x) \) starts from \( -2 \) at \( x = 0 \), reaches \( 2 \) at \( x = 1 \), and continues to increase.

### Step 5: Find Intersection Points
To find the number of real solutions, we need to determine how many times the graphs of \( f_1(x) \) and \( f_2(x) \) intersect.

From the values calculated:
- At \( x = 0 \): \( f_1(0) = 1 \) and \( f_2(0) = -2 \) (no intersection)
- At \( x = 1 \): \( f_1(1) = 2 \) and \( f_2(1) = 2 \) (intersection at this point)
- As \( x \) increases beyond \( 1 \), \( f_1(x) \) continues to increase and \( f_2(x) \) also increases, indicating another intersection.

### Conclusion
The two functions intersect at two points, thus the total number of real solutions to the equation \( x^3 + 1 = 2 \sqrt[3]{2x - 1} \) is **2**.