Question

Hello! I need help on the following problem. If possible, please write explanations out in words so I can fully understand all the steps required. Thank you!

Let \(g(x)\) be a function defined on \(\mathbb{R}\) such that \(g^{\prime \prime}(x)=\frac{1}{1+x^{2}}, g(0)=0\) and \(g^{\prime}(0)=0\).

(a) What is the range of \(g^{\prime \prime}(x)\) ?

(b) Use part \((a)\) to show that \(g^{\prime}(x)\) is increasing.

(c) Use the Mean Value Theorem and part ( \(a\) ) to show that for any \(x \geq 0\),

$$ g^{\prime}(x) \leq x . $$

(d) Find all critical points of \(g\) and classify each as a local max, a local min or neither. Be sure to explain how you know there are not any other critical points apart from the ones you find.

(e) Find the absolute extrema of \(g\). If they don't exist, explain why.

(f) Bonus: Use the Mean Value Theorem and part \((c)\) to show that for any \(x \geq 0\),

$$ g(x) \leq x^{2} $$

Answer 1