TIFR CSE 2020 | Part A | Question: 8

Question

Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly one global minimum inside $(0,1)$. What can you say about the behaviour of the first derivative $f’$ and and second derivative $f’’$ on $(0,1)$ (give the most precise answer)?

• A. $f’$ is zero at exactly two points, $f’’$ need not be zero anywhere
• B. $f’$ is zero at exactly two points, $f’’$ is zero at exactly one point
• C. $f’$ is zero at  at least two points, $f’’$ is zero at exactly one point
• D. $f’$ is zero at at least two points, $f’’$ is zero at at least ​​​​ one point
• E. $f’$ is zero at at least two points, $f’’$ is zero at at least two points

Comments

If f'(×) is zero then f''(x) will be also zero right so when f'(x) is zero at atleast two points then f''(xl is also zero at atleast two points right then E will be ans righr

Answer 1

Function $f$ is differentiable in domain $[0,1],$ it means $f$ is continuous in the given domain and no corner edge is present.

Since, $f$ has exactly one global maxima and one global minima but it can have many local maxima/minima, So, $f'$ must be zero at atleast $2$ points which is shown below.

Now, possible options are : $b,c,d.$

Now, consider some function $f(x)$ as given below which eliminates option $(e)$  :

Now, consider some function $f(x)$ as given below which eliminates option $(c)$  :

Answer 2

Since the function f(x) has exactly one global minima and one global maxima, f’(x) is guaranteed to be zero on those x.

Also, since f’(x) is zero for two distinct points, we can apply rolle’s theorem since f’(x0) = f’(x1) = 0  ( x0,x1 are points of global maxima and minima).

So, we could find a point d in the interval [x0,x1] or [x1,x0] such that f’’(d) = 0.

Rolle’s theorem only guarantees a point and not the number of such points. So, it could be more than one too.