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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

Let fx denote the function graphed Estimate where fxa has local minimab has local maximac has inflection pointsd is increasinge is decreasingf is concave upg is class=

Respuesta :

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)

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