Question

Given that $f ( x )$ is continuous and differentiable function on $a \leq x \leq b$ where $a < b , f ( a )<0$ and $f(b)>0$, which of the following are always true?
(i) $ f ( x )$ is bounded on $a \leq x \leq b$.
(ii) The equation $f ( x )=0$ has at least one solution in $a < x < b$.
(iii) The maximum and minimum values of $f ( x )$ on $a \leq x \leq$ b occur at points where $f ^{\prime}( c )=0$.
(iv) There is at least one point c with a $< c <$ b where $f ^{\prime}$ (c) $>0$.
(v) There is at least one point $d$ with $a < d <$ b where $f ^{\prime}$ (d) $<0$.

• A. Only (ii) and (iv) are true
• B. all except (iii) are true
• C. all except (v) are true
• D. only (i), (ii) and (iv) are true

Answer 1

Answer: (D)

(i) This statement is true, every continuous function is bounded on a closed interval
(ii) True again, by Intermediate Value Theorem
(iii) Not true, because maximum and/or minimum values could also occur at a or $b$.
(iv) True. By the Mean Value Theorem there exist a point between a and b where the derivative is exactly $\frac{ f ( b )- f ( a )}{ b - a }$ a clearly positive value.
(v) Not always true, for example the function might be strictly increasing guaranteeing the derivative to be always positive.
Thus the true statements are (i), (ii) and (iv) and the correct answer is (D)