Second derivative：練習問題

Given $$g(x) = \ln(4x)$$, find $$g''(x)$$.

Find the second derivative of $$h(x) = x^2 \ln x$$.

Let \(f(x) = x^n e^{-x}\), where \(n\) is a positive integer. If \(f''(2) = 0\), find all possible value(s) of \(n\).

Find the second derivative of the function $$f(x) = 2x^5 - 3x^4 + 6x^2 - 1$$.

If $$y = 5e^{3x}$$, determine $$\frac{d^2y}{dx^2}$$

Consider the function \(g(x) = x^2 \ln x\) for \(x > 0\).
Find the second derivative \(g''(x)\).

For the function \(f(x) = x \ln x\) where \(x > 0\), find \(f''(x)\) and determine the concavity of the function at \(x=e^2\).

Consider the function \(f(x) = \frac{\ln x}{x^2}\) for \(x > 0\).
(a) Find \(f''(x)\).
(b) Find the exact coordinates of the point of inflection of the graph of \(y = f(x)\) and determine the interval(s) of \(x\) where the graph is concave downwards.

Given the function \(f(x) = x^3 - 6x^2 + 5x - 1\).

(a) Find \(f'(x)\) and \(f''(x)\).

(b) Determine the interval(s) where the graph of \(f(x)\) is concave up.

Let \( f(x) = (x^2 + 1)e^{-x} \) for all real \( x \).

(a) Find \( f'(x) \) and \( f''(x) \). (3 marks)

(b) Determine whether the curve \( y = f(x) \) has a local extremum at \( x = 1 \). Justify your answer using the second derivative test. (3 marks)

(c) Find the interval(s) where the curve \( y = f(x) \) is concave upward. (2 marks)