Let \(\displaystyle f(x)=C_1 \cos x+ C_2\sin x\). \(\displaystyle f''(x)+f(x)=?\)

Difficulty: 2.

Calculate the following derivatives:

 (1)  \(\displaystyle \big(e^{(x^3)}\big){}^{(60)}(0)\),      (2)  \(\displaystyle \big( e^{x^4}\big)^{(102)}(0)\),      (3)  \(\displaystyle \big( e^{x^4}\big)^{(100)}(0)\).

Difficulty: 2.

How many times the function \(\displaystyle |x|^3\) is differentiable at \(\displaystyle 0\)?

Difficulty: 3.

Is it true that if \(\displaystyle f\) is 7 times differentiable on \(\displaystyle \R\), \(\displaystyle \lim_{x\to-\infty}f(x)=5\) and \(\displaystyle \lim_{x\to\infty}f(x)=3\), then \(\displaystyle f\) has an inflection point?

Difficulty: 4.

Find a function which is \(\displaystyle k\) times differentiable at \(\displaystyle 0\) but not \(\displaystyle k+1\) times.

Difficulty: 4.

How many times the function \(\displaystyle |x|^\alpha\) is differentiable at \(\displaystyle 0\) if \(\displaystyle \alpha>0\)?

Difficulty: 4.

Is it true that if \(\displaystyle f'''(x)=f(x)\) for all \(\displaystyle x\in\R\)-re, then \(\displaystyle f(x)=c\cdot e^x\) for some \(\displaystyle c\in\R\)?

Difficulty: 5.

Calculate all derivatives of

\(\displaystyle f(x)=\frac{ax+b}{cx+ d}.\)

Difficulty: 5.

Assume that \(\displaystyle f\in C^{\infty}(0,\infty)\), \(\displaystyle \lim_{0+0}f=\lim_{\infty} f=0\). Prove that \(\displaystyle \exists \xi>0\): \(\displaystyle f^{''}(\xi)=0\).

Difficulty: 5.

Assume that \(\displaystyle f\) and \(\displaystyle g\) are \(\displaystyle n\) times differentiable at the point \(\displaystyle a\).

(a) Prove that \(\displaystyle fg\) is also \(\displaystyle n\) times differentiable at the point \(\displaystyle a\).

(b) \(\displaystyle (fg)^{(n)}(a)=?\)

Difficulty: 5.

Prove that

\(\displaystyle (1-x^2)T_n''(x)-xT_n'(x)+n^2T_n(x)=0.\)

Difficulty: 5.

Is it true that if \(\displaystyle f\) is 2 times differentiable at \(\displaystyle a\), then

\(\displaystyle \lim_{h\to 0}\frac{f(a+2h)-2f(a+h)+f(a)}{h^2} = f''(a)\ ?\)

Difficulty: 6. Hint is provided for this problem.

Find a differentiable function \(\displaystyle f\) which is equal to \(\displaystyle 2x\) for \(\displaystyle x\leq 0\), and equal to \(\displaystyle 3x\) for \(\displaystyle x\geq 1\). Is there a 2 times differentiable function? And a 271 times differentiable function?

Difficulty: 6.