91Ó°ÊÓ

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f\) is concave down and \(f(x)\) is positive for all \(x\).

Which of the limits cannot be computed with I'Hopital's rule? (a) \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) (b) \(\lim _{x \rightarrow 0} \frac{\cos x}{x}\) (c) \(\lim _{x \rightarrow 0} \frac{x}{\sin x}\) (d) \(\lim _{x \rightarrow \infty} \frac{x}{e^{x}}\)

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f\) is concave down and \(f(x)\) is negative for all \(x\).

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f\) is concave up and \(f(x)\) is negative for all \(x\).

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f(x) f^{\prime \prime}(x)<0\) for all \(x\).

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f(x) f^{\prime}(x) f^{\prime \prime}(x) f^{\prime \prime \prime}(x) < 0\) for all \(x\).

Given that \(f^{\prime}(x)\) is continuous everywhere and changes from negative to positive at \(x=a,\) which of the following statements must be true? (a) \(a\) is a critical point of \(f(x)\) (b) \(f(a)\) is a local maximum (c) \(f(a)\) is a local minimum (d) \(f^{\prime}(a)\) is a local maximum (e) \(f^{\prime}(a)\) is a local minimum