We say an equation `f(x)=g(x)` is consistent, if the curves `y=f(x) and y=g(x)` touch or intersect at atleast one point. If the curves `y=f(x) and y=g(x)` do not intersect or touch, then the equation `f(x)=g(x)` is said to be inconsistent i.e. has no solution.
The equation `sinx=x^(2)+x+1` is

Options

A. consistent and has infinite number of solutions
B. consistent and has finite number of solutions
C. inconsistent
D. None of the above

Answer

To determine whether the equation \( \sin x = x^2 + x + 1 \) is consistent or inconsistent, we will analyze the functions \( f(x) = \sin x \) and \( g(x) = x^2 + x + 1 \).

### Step 1: Analyze the function \( g(x) = x^2 + x + 1 \)

1. **Identify the nature of the quadratic function**:
- The quadratic function \( g(x) = x^2 + x + 1 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive).
- To find the vertex of the parabola, we can use the vertex formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 1 \):
\[ x = -\frac{1}{2 \cdot 1} = -\frac{1}{2} \]
- Now, substituting \( x = -\frac{1}{2} \) into \( g(x) \) to find the minimum value:
\[ g\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{3}{4} \]
- Thus, the minimum value of \( g(x) \) is \( \frac{3}{4} \), which occurs at \( x = -\frac{1}{2} \).

### Step 2: Analyze the function \( f(x) = \sin x \)

2. **Identify the range of the sine function**:
- The function \( f(x) = \sin x \) oscillates between -1 and 1 for all \( x \).
- Therefore, the maximum value of \( \sin x \) is 1 and the minimum value is -1.

### Step 3: Compare the two functions

3. **Determine if the two functions can intersect**:
- Since the minimum value of \( g(x) \) is \( \frac{3}{4} \) and the maximum value of \( f(x) \) is 1, we need to check if \( g(x) \) can equal \( f(x) \).
- The function \( g(x) \) is always greater than or equal to \( \frac{3}{4} \) for all \( x \).
- The function \( f(x) \) can take values from -1 to 1, but it can never reach \( \frac{3}{4} \) or exceed it.

### Step 4: Conclusion

4. **Final determination**:
- Since \( g(x) \) is always greater than or equal to \( \frac{3}{4} \) and \( f(x) \) can take values only up to 1, there are no points where \( \sin x \) can equal \( x^2 + x + 1 \).
- Therefore, the equation \( \sin x = x^2 + x + 1 \) is inconsistent, meaning it has no solution.

### Final Answer:
The equation \( \sin x = x^2 + x + 1 \) is inconsistent.

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