Practice Problems for 4/8
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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?
- \(c\) is always an inflection point of \(f(x)\).
- \(c\) is only an inflection point of \(f(x)\) if \(f'(c)\) is a local maximum or local minimum value of \(f'(x)\).
- \(c\) is only an inflection point of \(f(x)\) if \(f(c)\) is a local maximum or local minimum value of \(f(x)\).
- 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.
- \(f(x)=x^4-4x^3+10\)
- \(f(x)=\frac{1}{12}x^4-\frac{1}{3}x^3+\frac{1}{2}x^2+x\)
- Graph the function \(f(x)=x^2+\frac{2}{x}\).