## More Problems on the Derivative

In these pages we discuss some problems related to the derivative concept. Most of the problems are challenging and not easy. We will give the answer to some of them The readers are encouraged to find the solutions for the rest. Send it and we will publish it. Enjoy.

**Problem 1.** Show that the function

is differentiable at 0 at any order. Find
*f*^{(n)}(0),

**Answer.**

**Problem 2.** Let *f*(*x*) be a function defined on (-*a*,*a*), with *a* > 0. Assume that *f*(*x*) is continuous at 0 and

where *k* satisfies 0 < *k* < 1.
- (a)
- Show that
*f*(*x*) is differentiable at 0 and that
.
- (b)
- Show that the conclusion obtained in (a) is still valid even when
*k* > 1.

**Answer.**

**Problem 3.** For any
,
set

- (a)
- Show that
is differentiable.
- (b)
- Is
continuous at 0?
- (c)
- Show that if
,
then
is not monotone on any open interval which contains 0.

**Answer.**

**Problem 4.** Let *f*(*x*) be a differentiable function. Assume that *f*'(*x*) is continuous. Define the sequence
by

Assume that
is not almost constant. Show that if
converges to ,
then

**Answer.**

**Problem 5.** Consider the function
*f*(*x*) = (*x*^{2}-1)^{n}, for .
Show that

**Answer.**

**Problem 6.** Consider the function
.
Show that

Then show that
*f*^{(2n+1)}(0) = 0, for any .

**Answer.**

**Problem 7.** Let
,
where
.
Let *a* and *b* be such that

Show that there exists
such that

Find *c*.

**Answer.**

**Problem 8.** Show that *e*^{x} = 1-*x* has a unique solution. Then find that solution.

**Answer.**

**Problem 9.** Show that for any
,
we have

**Answer.**

**Problem 10.** Show that for any *a* and *b* such that 0 < *a*< *b*, we have

**Answer.**

**Problem 11.** Let
.

- (1)
- Show that the equation
has one solution
*x*_{0} in the interval
.
- (2)
- Show that there exists
such that

**Answer.**

**
**

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