Which of the following graphs represent functions whose derivatives have a maximum in the interval $(0, 1)$ ?
My reasoning goes like this:
From the graph sketches, we can readily conclude, that all of the $4$ graphs are continuous in the interval $(0,1).$ Now, we use the following theorem on continuity:
- The function $f(x)$ continuous on the the interval $[a,b]$, possesses the following properties:
$(1)$ $f(x)$ is bounded on $[a,b]$
$(2)$ $f(x)$ has the minimum and maximum values on $[a,b]$
$(3)$ If $m=\text{min}_{a\leq x\leq b} f(x)$ and $M=\text{max}_{a\leq x\leq b} f(x),$ then for any $A$ satisfying the inequalities $m\leq A\leq M,$ $\exists$ a point $x_0\in [a,b]$ for which $f(x_0)=A$
- In particular , if a function is defined and continuous in some interval, and if this interval is not a closed one, then it can neither have the greatest nor the least value.
In all these graphs, it is seen that, in the interval $(0,1)$ all the 4 functions sketched, are continuous. Since, $(0,1)$ is an open interval and from the above theorem, neither of the functions do not have the greatest nor the least value in this interval.
In option $A$ we see, that $f(x)$ is monotonically decreasing, and hence, $f'(x)<0,$ and since $f$ has no maxima and minima in the open interval, $f'(x)$ is always, negative and itself a decreasing function. So, $f'(x)$ never attains a maximum value, since there is no greatest value of $f(x)$ nor any least value of $f(x)$.
In option $B,$ it's a monotonically increasing function, and $f'(x)>0$. But by analogous reasoning as above there is no, maximum value of $f'(x)$.
In case, of the last two options, by similar reasoning, $f'(x)$ has no maximum value.
However, I feel that something is wrong with my solution. I don't know what it is. Is the above solution correct? (As neither of the options seems correct, according to my solution ) If not, where is it going wrong? I am not quite getting it.
A thing to add: Is there a general way of solving these sort of problems. I am anticipating, a preferable way, of solving these sort of problems, in an mcq(Multiple Choice Question) exam, where one has a time constraint.
I know, that some users don't advocate the use of pictures, but the graphs are part of this problem and so, I attached them, as seen in the beginning of this post.
Does it have a maximum value or maximum absolute value? If it's maximum value it's the steepest uphill slope (since the steeper the slope, the larger the value of the derivative), if it's maximum absolute value it's the steepest slope which can be either uphill or downhill. It's either B or D since A and C have functions that get steeper outside of that interval.