Use the given graph of f to find the following

Question

http://imgur.com/2dZt89y

Use the given graph of f to find the following

(a) The open intervals on which f is increasing.

(b) The open intervals on which f is decreasing.

(c) The numbers x where f 0 (x) = 0.

(d) The critical numbers of f (excluding the end points).

(e) The open intervals on which f is concave upward.

(f) The open intervals on which f is concave downward.

(g) The coordinates of the points of inflection.

Answer 1

(a)

Step 1 :

A function is said to be "increasing" when the y - value increases as the x - value increases.

Now find the intervals of increasing by observing the graph :

The graph is increasing over the intervals (1, 3) and (4, 6).

(b)

Step 2 :

A function is said to be "decreasing" when the y - value decreases as the x - value increases.

Now find the intervals of decreasing by observing the graph :

The graph is decreasing over the intervals (0, 1) and (3, 4).

(c)

Step 3 :

Find the points where the function f(x) = 0.

f(x) = 0 means the points where the intersects the x - axis.

Now observe the graph, graph no where intersecting the x - axis.

So no x - intercepts.

Solution :

(a) The graph is increasing over the intervals (1, 3) and (4, 6).

(b) The graph is decreasing over the intervals (0, 1) and (3, 4).

(c) No x - intercepts. ( No numbers on x where f(x) = 0).

Answer 2

(d)

Step 1 :

Definition of Critical point :

The graph of the function at which it has a cusp with vertical tangent.

Now find the critical point by observing the graph :

The graph has a critical numbers at x = 1, x = 3 and x = 4.

(e)

Step 2 :

Concave up :

A function is said to be concave upwards if it opens (bends) up.

Observe the graph to find the interval of concave upwards.

The interval of concave upwards is (0, 2).

(f)

Step 3 :

Concave down :

A function is said to be concave downwards if it opens (bends) down.

Observe the graph to find the interval of concave downwards.

The interval of concave downwards is (2, 4).

(g)

Step 4 :

Definition of an inflection point:

An inflection point occurs on f(x) at x₀ if and only if f(x) has a tangent line at x₀ and there exists and interval containing x₀ such that f(x) is concave up on one side of x₀ and concave down on the other side.

Now observe the graph to find inflection points.

The inflection point is (2, 3).

Solution :

(d) The critical numbers are x = 1, x = 3 and x = 4.

(e) The interval of concave upwards is (0, 2).

(f) The interval of concave downwards is (2, 4).

(g) The inflection point is (2, 3).