Question

Let $f$ be derivable in $[0, 1]$, then

• A. there exists c ε ( 0,1) such that ∫c,o f (x) dx (1-c) f(c)
• B. there does not exist any point d (0, 1) for which ∫d,0 f(x) dx = (1-d ) f(d)
• C. ∫c,0 f(x) dx does not exist , for any c ε (0,1)
• D. ∫c,0 f(x) dx is independent of c, cε (0,1)

Answer 1

The Correct Option is A

We are given that \( f \) is differentiable on the interval \([0, 1]\), and we need to analyze the given statements. Step 1: Consider the integral and the point \( c \) We are interested in the integral: \[ \int_0^c f(x) \, dx \] and we are asked whether there exists a point \( c \in (0, 1) \) such that the following relationship holds: \[ \int_0^c f(x) \, dx = (1 - c) f(c) \] This question is related to the application of the **Mean Value Theorem for Integrals**. The Mean Value Theorem for Integrals states that if \( f \) is continuous on \([a, b]\), then there exists a point \( c \in [a, b] \) such that: \[ \int_a^b f(x) \, dx = f(c)(b - a) \] Here, we can apply this theorem to the interval \([0, c]\), which gives us: \[ \int_0^c f(x) \, dx = f(c) \cdot c \] So, the equation we are checking becomes: \[ f(c) \cdot c = (1 - c) \cdot f(c) \] Simplifying this, we get: \[ c = (1 - c) \] which implies: \[ c = \frac{1}{2} \] Thus, there exists a point \( c \in (0, 1) \) where the integral satisfies the given relationship. Therefore, the first statement is correct. Step 2: Analyze the other options - The second option says that there does not exist any point \( d \in (0, 1) \) for which the relationship holds. This is incorrect, as we showed that \( c = \frac{1}{2} \) satisfies the equation. - The third option states that \( \int_0^c f(x) \, dx \) does not exist for any \( c \in (0, 1) \). This is not true, as the integral is well-defined for any differentiable function \( f \) on \([0, 1]\). - The fourth option suggests that \( \int_0^c f(x) \, dx \) is independent of \( c \), which is clearly incorrect since the value of the integral depends on the upper limit of integration, \( c \). Conclusion The correct answer is: \[ \boxed{\text{There exists } c \in (0, 1) \text{ such that } \int_0^c f(x) \, dx = (1 - c) f(c)} \]