To tackle the problem involving the differentiable function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the equation \( 2f\left(x + \frac{y}{2}\right) - f(y) = f(x) \), along with the conditions \( f(0) = 5 \) and \( f'(5) = -1 \), we need to analyze the implications of the functional equation and the given conditions to evaluate the options provided.
Understanding the Functional Equation
The equation \( 2f\left(x + \frac{y}{2}\right) - f(y) = f(x) \) suggests a relationship between the values of the function at different points. To simplify our analysis, we can substitute specific values for \( x \) and \( y \). Let's start with \( y = 0 \):
- Substituting \( y = 0 \) gives us \( 2f\left(x + 0\right) - f(0) = f(x) \), which simplifies to \( 2f(x) - 5 = f(x) \). This leads to \( f(x) = 5 \) for all \( x \).
This indicates that \( f(x) \) could be a constant function. However, we also have \( f'(5) = -1 \), which contradicts the idea of \( f(x) \) being constant since the derivative of a constant function is zero. Therefore, we need to explore further.
Exploring the Derivative
Next, let's differentiate the functional equation with respect to \( y \) to gain more insights:
- Taking the derivative gives us \( 2f'\left(x + \frac{y}{2}\right) \cdot \frac{1}{2} - f'(y) = 0 \). This simplifies to \( f'\left(x + \frac{y}{2}\right) = f'(y) \).
This implies that \( f' \) is constant, which means \( f(x) \) is a linear function. Let’s denote it as \( f(x) = mx + b \). Given \( f(0) = 5 \), we find \( b = 5 \), so \( f(x) = mx + 5 \).
Finding the Slope
We also know \( f'(x) = m \). Since \( f'(5) = -1 \), we have \( m = -1 \). Thus, the function can be expressed as:
f(x) = -x + 5
Evaluating the Options
Now that we have determined the function, we can evaluate the options provided:
- A) lim (f(x))^1/4 = e as x approaches 4: We calculate \( f(4) = -4 + 5 = 1 \). Thus, \( \lim_{x \to 4} (f(x))^{1/4} = 1^{1/4} = 1 \), which is not equal to \( e \). This option is false.
- B) f(|x|) is non-derivable at exactly 2 points: The function \( f(x) = -x + 5 \) is linear, and thus \( f(|x|) \) is non-derivable at \( x = 0 \) (due to the absolute value function). Therefore, this option is true.
- C) Area bounded by f(x), x-axis, and y-axis is equal to 25 sq units: The area under the line from \( x = 0 \) to \( x = 5 \) is a triangle with base 5 and height 5, giving an area of \( \frac{1}{2} \times 5 \times 5 = 12.5 \), which is not equal to 25. This option is false.
- D) |f(|x|)| is non-derivable at exactly 3 points: Since \( f(|x|) \) is non-derivable at \( x = 0 \) and remains linear elsewhere, this option is also false.
Final Assessment
Based on our analysis, the only true statement is option B, which asserts that \( f(|x|) \) is non-derivable at exactly 2 points. The other options do not hold true based on the properties of the function we derived.
