Question

Let f(x) denote the function graphed. Estimate where f(x)(a) has local minima(b) has local maxima(c) has inflection points(d) is increasing(e) is decreasing(f) is concave up(g) is concave down

Answer 1

We must find some properties of the function f(x) plotted in the graph.

(a) Local minima

We see that the function has a local minimum at the point (0,0).

(b) Local maxima

We see that the function has a local maximum at the point (-2,4).

(c) Inflection points

By definition, an inflexion point of a curve is a point at which a change in the direction of curvature occurs.

We see that in the interval (-2, -1) the slope of the curve decreases, but at the point (-1, 2) we have a change of the curvature and the slope starts to increase. So (-1, 2) is an inflexion point.

(d) Is increasing

The function is increasing when the slope of a tangent line to the curve is positive.

We see that the function has a tangent line with a positive slope on the intervals:

• (-∞, -2),

,

• (0, ∞).

So the function is increasing in the interval (-∞, -2) U (0, ∞).

(e) Is decreasing

The function is increasing when the slope of a tangent line to the curve is negative.

We see that the function has a tangent line with a negative slope on the intervals:

• (-2, 0).

So the function is increasing in the interval (-2, 0).

(f) Concave up

The curve is concave up in the zones with a U shape.

So the curve is concave up in the interval (-1, ∞).

(g) Concave down

The curve is concave down in the zones with a ∩ shape.

So the curve is concave down in the interval (-∞, -1).

Answers

(a) Point (0, 0)

(b) Point (-2, 4)

(c) Point (-1, 2)

(d) Interval (-∞, -2) U (0, ∞)

(e) Interval (-2, 0)

(f) Interval (-1, ∞)

(g) Interval (-∞, -1)