mathematicalcircles
To Live and Breathe Mathematics
Friday 20 August 2021
Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by
\[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{cases}\]
Then,\(\qquad\)
(a) \(f\) is discontinuous.
\(\qquad\)
(b) \(f\) is continuous but not differentiable.
\(\qquad\) \(\qquad\) \(\qquad\)
(c) \(f\) is differentiable and its derivative is discontinuous.
\(\qquad\) \(\qquad\) \(\qquad\)\(\qquad\)\(\qquad\) \(\qquad\)
(d) \(f\) is differentiable and its derivative is continuous.
Wednesday 18 August 2021
Let
\[\begin{gathered} p(x)=x^{3}-3 x^{2}+2 x, x \in \mathbb{R}, \\ f_{0}(x)= \begin{cases}\int_{0}^{x} p(t) d t, & x \geq 0, \\ -\int_{x}^{0} p(t) d t, & x<0,\end{cases} \\ f_{1}(x)=e^{f_{0}(x)}, \quad f_{2}(x)=e^{f_{1}(x)}, \quad \ldots \quad, f_{n}(x)=e^{f_{n-1}(x)} \end{gathered}\]
How many roots does the equation \(\frac{d f_{n}(x)}{d x}=0\) have in the interval \((-\infty, \infty)?\)
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Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, &a...
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Find the number of pairs \((m,n)\) of positive integers with \(1 \le m < n \le 30\) such that there exists a real number \(x\) satisfying...