4.5 Derivatives and the shape of a graph  (Page 7/14)

a. Increasing over $-\frac{\pi }{2}<x<\frac{\pi }{2},$ decreasing over $x<-\frac{\pi }{2},x>\frac{\pi }{2}$ b. Local maximum at $x=\frac{\pi }{2};$ local minimum at $x=-\frac{\pi }{2}$

a. Concave up for $x>\frac{4}{3},$ concave down for $x<\frac{4}{3}$ b. Inflection point at $x=\frac{4}{3}$

a. Increasing over $x<0$ and $x>4,$ decreasing over $0<x<4$ b. Maximum at $x=0,$ minimum at $x=4$ c. Concave up for $x>2,$ concave down for $x<2$ d. Infection point at $x=2$

a. Increasing over $x<0$ and $x>\frac{60}{11},$ decreasing over $0<x<\frac{60}{11}$ b. Minimum at $x=\frac{60}{11}$ c. Concave down for $x<\frac{54}{11},$ concave up for $x>\frac{54}{11}$ d. Inflection point at $x=\frac{54}{11}$

a. Increasing over $x>-\frac{1}{2},$ decreasing over $x<-\frac{1}{2}$ b. Minimum at $x=-\frac{1}{2}$ c. Concave up for all $x$ d. No inflection points

a. Increases over $-\frac{1}{4}<x<\frac{3}{4},$ decreases over $x>\frac{3}{4}$ and $x<-\frac{1}{4}$ b. Minimum at $x=-\frac{1}{4},$ maximum at $x=\frac{3}{4}$ c. Concave up for $-\frac{3}{4}<x<\frac{1}{4},$ concave down for $x<-\frac{3}{4}$ and $x>\frac{1}{4}$ d. Inflection points at $x=-\frac{3}{4},x=\frac{1}{4}$

a. Increasing for all $x$ b. No local minimum or maximum c. Concave up for $x>0,$ concave down for $x<0$ d. Inflection point at $x=0$

a. Increasing for all $x$ where defined b. No local minima or maxima c. Concave up for $x<1;$ concave down for $x>1$ d. No inflection points in domain

a. Increasing over $-\frac{\pi }{4}<x<\frac{3\pi }{4},$ decreasing over $x>\frac{3\pi }{4},x<-\frac{\pi }{4}$ b. Minimum at $x=-\frac{\pi }{4},$ maximum at $x=\frac{3\pi }{4}$ c. Concave up for $-\frac{\pi }{2}<x<\frac{\pi }{2},$ concave down for $x<-\frac{\pi }{2},x>\frac{\pi }{2}$ d. Infection points at $x=\text{±}\frac{\pi }{2}$

a. Increasing over $x>4,$ decreasing over $0<x<4$ b. Minimum at $x=4$ c. Concave up for $0<x<8\sqrt[3]{2},$ concave down for $x>8\sqrt[3]{2}$ d. Inflection point at $x=8\sqrt[3]{2}$

$f>0,f^{\prime }>0,f\text{″}<0$

$f>0,f^{\prime }<0,f\text{″}<0$

$f>0,f^{\prime }>0,f\text{″}>0$

True, by the Mean Value Theorem

True, examine derivative