18 thoughts on “Challenge Problems

  1. Prove or Disprove the following statement:
    $$
    \int 2 \sin x \cos x \; dx= -\frac{\cos{2x}}{2}+C_1
    $$
    and
    $$
    \int 2 \sin x \cos x \; dx= \sin^2{x} + C_2
    $$

  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?

  3. Prove/Disprove: If \(f'(x) >0 \) for \( x < a \)  and \( f'(x) < 0 \) for \( x > a \), the \(x = a\) is a relative maximum.

  4. Prove/Disprove: If \(x = a\) is a vertical asymptote of \(f(x) \), then \(f(x)\) is undefined at \(x = a\).

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