Anyone is allowed to participate. Justify your answers.
18 thoughts on “Challenge Problems”
Prove or Disprove the following statement:
$$
\int 2 \sin x \cos x \; dx= -\frac{\cos{2x}}{2}+C_1
$$
and
$$
\int 2 \sin x \cos x \; dx= \sin^2{x} + C_2
$$
Let \(f(x)= \sin x\). Does there exist an equation of a parabola: \(g(x) = ax^2 + bx + c\) (\(a \not = 0\)) such that for at least three values of \(x\), the function \(f\) and \(g\) agree?
Prove or Disprove the following statement:
$$
\int 2 \sin x \cos x \; dx= -\frac{\cos{2x}}{2}+C_1
$$
and
$$
\int 2 \sin x \cos x \; dx= \sin^2{x} + C_2
$$
Let \(f(x)= \sin x\). Does there exist an equation of a parabola: \(g(x) = ax^2 + bx + c\) (\(a \not = 0\)) such that for at least three values of \(x\), the function \(f\) and \(g\) agree?
Prove/Disprove: If \(y = b \) is a horizontal asymptote then \(f(x) \not = b \) for any \(x\).
Does there exist a function which is both even and odd? If so, find all such functions.
This is probably not the solution you are looking for but, f(x) = 0 is both even and odd.
Is this the only one? If so, prove that there are no other functions that are both even and odd.
Prove/Disprove: If \(f'(x) >0 \) for \( x < a \) and \( f'(x) < 0 \) for \( x > a \), the \(x = a\) is a relative maximum.
Prove/Disprove: If \(x = a\) is a vertical asymptote of \(f(x) \), then \(f(x)\) is undefined at \(x = a\).