HELP ME PLEASE!!!

Question

Use the graph of f '

 to identify the critical numbers of f, identify the open intervals on which f is increasing or decreasing, and determine whether f has a relative maximum, a relative minimum, or neither at each critical number. (If an answer does not exist, enter DNE.)

(a)

(i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.)

x = ______

(ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer using interval notation.)

increasing ______

decreasing ______

(iii) Determine whether f has a relative maximum, a relative minimum, or neither at each critical number. (Enter your answers as a comma-separated list.)

relative minimum x =______

relative maximum x =______

b)

(i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.)

x = ______

(ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer using interval notation.)

increasing ______

decreasing ______

(iii) Determine whether f has a relative maximum, a relative minimum, or neither at each critical number. (Enter your answers as a comma-separated list.)

relative minimum x =______

relative maximum x =______

c)

(i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.)

x =______

(ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer using interval notation.)

increasing______

decreasing______

(iii) Determine whether f has a relative maximum, a relative minimum, or neither at each critical number. (Enter your answers as a comma-separated list.)

relative minimum x =______

relative maximum x =______

d)

(i) Identify the critical numbers of f. (Enter your answers as a comma-separated list.)

x =______

(ii) Identify the open interval(s) on which f is increasing or decreasing. (Enter your answer using interval notation.)

increasing______

decreasing______

(iii) Determine whether f has a relative maximum, a relative minimum, or neither at each critical number. (Enter your answers as a comma-separated list.)

relative minimum x =______

relative maximum x =______

Answer 1

a)

i) Critical points are when f'(x)=0, or visually, where the graph may "peak" or "dip" or otherwise change. Here, we see there are no points where the derivative equals 0, so x does not exist.

ii) We see from the graph that f(x) is increasing across the interval (-∞,+∞).

iii) As there are no critical points, there are no relative maxima or minima.

b)

i) Here, we do have a critical point. We can see that at x=3/2, the graph changes from decreasing to increasing.

ii) The graph appears to be decreasing from (-∞,2] and then increasing from [2,+∞)

iii) As we do have a well-defined point such that f'(x)=0, namely x=3/2, we will also have a relative maximum or minimum. Looking at the graph, we can see that this is a relative minimum.

c)

i) Here we have two critical numbers, located at x=1/2 and x=3/2

ii) Increasing: [1/2,3/2]

Decreasing: (-∞,1/2]∪[3/2,+∞]

iii) Relative minimum: x=1/2

Relative maximum: x=3/2

d)

i) x=-1/2, 1, 2

ii) Increasing: (-∞,-1/2]∪[1,2]

Decreasing:[-1/2,1]∪[2,∞]

iii)x has local maxima at x=-1/2,2

x has a local minimum at x=1