\( ( \frac { d z } { d x } ) ^ { 4 } + ( \frac { d ^ { 2 } z } { d x ^ { 2 } } ) ^ { 2 } = 0 \)

K For the following composite function, find an inner function \( u=g(x) \) and an outer function \( y=f(u) \) such that \( y=f(g(x)) \). Then calculate \( y=\tan 3 x^{5} \) Identify the inner and outer functions. Choose the correct answer. A. \( u=g(x)=x^{5} \) and \( y=f(u)=3 \tan u \) B. \( u=g(x)=\tan x \) and \( y=f(u)=3 u^{5} \) O. \( u=g(x)=\tan x^{5} \) and \( y=f(u)=3 u \) \( \frac{d y}{d x}=\square \)

Let \( f(x)=2 x^{3}+3 x^{2}-12 x+1 \). a. Find all points on the graph of \( f(x) \) at which the tangent line is horizontal. b. Find all points on the graph of \( f(x) \) at which the tangent line has slope 108 .

Find the intervals on which \( f \) is increasing and the intervals on which it is decreasing. \( f(x)=\frac{x^{4}}{4}-3 x^{3}+9 x^{2} \)

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. \( f(x)=x^{4}-3 x^{3}+9 \)

Given \( f(x, y)=3 x^{3}+x y^{2}-2 y^{6} \), find \( f_{y} \) \( 2 x-60 y \) \( 18 x \) \( 2 y \) None of these answers