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Practice Problems for 4/8

  1. Let \(f(x)\) be a twice differentiable function. If \(x=c\) is a critical point of the function \(f'(x)\), which of the follow is true and WHY?
    1. \(c\) is always an inflection point of \(f(x)\).
    2. \(c\) is only an inflection point of \(f(x)\) if \(f'(c)\) is a local maximum or local minimum value of \(f'(x)\).
    3. \(c\) is only an inflection point of \(f(x)\) if \(f(c)\) is a local maximum or local minimum value of \(f(x)\).

  2. Find all inflection points of \(f(x)\) and determine on which intervals \(f(x)\) is concave up, and on which intervals it is concave down.
    1. \(f(x)=x^4-4x^3+10\)
    2. \(f(x)=\frac{1}{12}x^4-\frac{1}{3}x^3+\frac{1}{2}x^2+x\)

  3. Graph the function \(f(x)=x^2+\frac{2}{x}\).