Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

3) If f is continuous on \(\left( {a,b} \right)\) ,then f attains an absolute maximum value \(f\left( c \right)\) and an absolute minimum value \(f\left( d \right)\) at some numbers c and d in \(\left( {a,b} \right)\).

Short answer

The given statement is false.

Step-by-step solution

Find \(f\left( x \right)\) is continuous

Here we take the counter-example and solve it, now.

let\(f:\left( {0,\infty } \right) \to R\)

\(f\left( x \right) = \frac{1}{x}\)

It is a rational function and \(x \ne 0\) in the given domain\(\left( {0,\infty } \right)\).

\(\therefore \) \(f\left( x \right)\) is continuous.

Find \(f\left( x \right)\) is absolute maxima

but\(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \frac{1}{x} = \infty \)

\(f\left( x \right)\) is no absolute maxima.

i.e there does not exist a positive number m.

such that,

\(\left| {f\left( x \right)} \right| \le m.\)

Hence our question statement is false.