Question

10. If $f \in C([0,1])$ and \begin{equation} \int_0^x f(t) dt = \int_x^1 f(t) dt \end{equation} for all $x \in [0, 1]$, show that $f(x) = 0$ for all $x \in [0,1]$.

          10. If $f \in C([0,1])$ and \begin{equation*} \int_0^x f(t) dt = \int_x^1 f(t) dt \end{equation*} for all $x \in [0, 1]$, show that $f(x) = 0$ for all $x \in [0,1]$.

10. If f ∈ C([0,1]) and
∫0^x f(t) dt = ^1 f(t) dt
for all x ∈ [0, 1], show that f(x) = 0 for all x ∈ [0,1].

Added by Lu-S B.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Adi S

Step 1

First, we know that f âˆˆ C([0, 1]), which means that f is a continuous function on the interval [0, 1]. Show more…

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If f âˆˆ C([0, 1]) and dt for all x âˆˆ [0,1], show that f(x) = 0 for all x âˆˆ [0,1]

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Question

10. If $f \in C([0,1])$ and \begin{equation*} \int_0^x f(t) dt = \int_x^1 f(t) dt \end{equation*} for all $x \in [0, 1]$, show that $f(x) = 0$ for all $x \in [0,1]$.

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10. If $f \in C([0,1])$ and \begin{equation} \int_0^x f(t) dt = \int_x^1 f(t) dt \end{equation} for all $x \in [0, 1]$, show that $f(x) = 0$ for all $x \in [0,1]$.