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Chapter 5: Problem 10

If \(f\) is continuous on \([a, b]\) and \(\int_{a}^{b}|f(x)| d x=0,\) what can you conclude about \(f ?\)

Short Answer

Expert verified
Answer: If the integral of the absolute value of a continuous function \(f\) on the interval \([a, b]\) is 0, it implies that the function \(f\) must be the zero function on the interval \([a, b]\).

Step by step solution

01

Recall the definition of the absolute value of a function

For any function \(f(x)\), its absolute value \(|f(x)|\) is defined as: $$ |f(x)| = \begin{cases} f(x) & \text{if } f(x) \geq 0, \\ -f(x) & \text{if } f(x) < 0. \end{cases} $$
02

Note the properties of the absolute value function

Remember that the absolute value function is always non-negative, i.e., for any \(x\) in its domain, \(|f(x)|\geq0\). This is important because it sets a lower bound to the value that the integral of \(|f(x)|\) can take.
03

Draw a conclusion about the integral given

Since \(|f(x)| \geq 0\) for all \(x\) in \([a, b]\) and the integral \(\int_{a}^{b}|f(x)| dx = 0\), this implies that the absolute value of the function \(f\) must be identically zero on the interval \([a, b]\). So, for all \(x\) in \([a, b]\), \(|f(x)|=0\).
04

Relate the absolute value to the original function

Since \(|f(x)|=0\) for all \(x\) in \([a, b]\), and by the definition of the absolute value, this implies that \(f(x)=0\) for all \(x\) in \([a, b]\). Therefore, we can conclude that \(f\) must be the zero function on the interval \([a, b]\).

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