Question

Given that $f ( x )$ is continuously differentiable 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 $f ^{\prime}$ (c) $<0$.

• A. only (ii) and (iv) are true
• B. all but (iii) are true
• C. all but (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, without the derivatives being 0 .
(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 guarenteeing the derivative to be always positive.
Thus the true statements are (i), (ii) and (iv) and the correct answer is (D)