Let f be a continuous function on the closed interval [0,1]. Let f(0)=1 and f(1)=0. Which of the following statement is not necessarily true?

Question

Let f be a continuous function on the closed interval [0,1]. Let f(0)=1 and f(1)=0. Which of the following statement is not necessarily true? (A) There exists a number k in [0,1] such that f ( k ) ≥ f ( x ) for all x in [0,1] (B) For all a and b in [0,1], if a=b, then f(a)=f(b) (C) There exists a number k in [0,1] such that f ( k )=(1/2) (D) There exists a number k in [0,1] such that f ( k )=(3/2)

Tardigrade Admin · Accepted Answer

Read the statement recall the properties of continuous functions.

Q. Let $f$ be a continuous function on the closed interval $[0, 1]$ . Let $f (0) = 1$ and $f (1) = 0$ . Which of the following statement is not necessarily true?

There exists a number $k$ in $[0, 1]$ such that $f (k) \geq f (x)$ for all $x$ in $[0, 1]$

For all $a$ and $b$ in $[0, 1]$ , if $a = b$ , then $f (a) = f (b)$

There exists a number $k$ in $[0, 1]$ such that $f (k) = \frac{1}{2}$

There exists a number $k$ in $[0, 1]$ such that $f (k) = \frac{3}{2}$

Read the statement
recall the properties of continuous functions.

Q. Let f be a continuous function on the closed interval [0,1]. Let f(0)=1 and f(1)=0. Which of the following statement is not necessarily true?

There exists a number k in [0,1] such that f(k)≥f(x) for all x in [0,1]

For all a and b in [0,1], if a=b, then f(a)=f(b)

There exists a number k in [0,1] such that f(k)=21​

There exists a number k in [0,1] such that f(k)=23​