The function `f(x)=x^(2), x in R` has no minimum value. (True/False)

Answer

To determine whether the statement "The function \( f(x) = x^2 \), \( x \in \mathbb{R} \) has no minimum value" is true or false, we will analyze the function step by step.

### Step 1: Understanding the function
The function given is \( f(x) = x^2 \). This is a quadratic function, which is a standard form of a parabola that opens upwards.

### Step 2: Finding the critical points
To find the minimum or maximum values of the function, we first need to find its derivative and set it to zero.

1. **Differentiate the function**:
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]

2. **Set the derivative to zero**:
\[ 2x = 0 \implies x = 0 \]

### Step 3: Determine if it is a minimum or maximum
Next, we will use the second derivative test to determine whether this critical point is a minimum or maximum.

1. **Find the second derivative**:
\[ f''(x) = \frac{d^2}{dx^2}(x^2) = 2 \]

2. **Evaluate the second derivative at the critical point**:
Since \( f''(x) = 2 \) is positive, this indicates that the function is concave up at \( x = 0 \).

### Step 4: Conclusion about the minimum value
Since \( f''(0) > 0 \), the critical point \( x = 0 \) is a minimum point. Now, we can find the minimum value of the function:

\[ f(0) = 0^2 = 0 \]

### Final Answer
The function \( f(x) = x^2 \) does have a minimum value, which is \( 0 \) at \( x = 0 \). Therefore, the statement "The function \( f(x) = x^2 \), \( x \in \mathbb{R} \) has no minimum value" is **False**.