# `f(x) = (x^3)(x - 2)^4` (a) Use a graph of `f` to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of `f''` to give better...

`f(x) = (x^3)(x - 2)^4` (a) Use a graph of `f` to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of `f''` to give better estimates.

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Using the graph of f(x), we determine the intervals of concavity based on the slopes of tangent line.

A** decreasing slopes** **of tangent** as we move to left to right (as x increase) indicates that the graph is **concave down.**

An **increasing slopes of tangent** as we move to left to right (as x increase) indicates that the graph is **concave up**

** Slope of tangent line =0** ** at critical points** (minimum, maximum or inflection point). Location of the critical point can be use as boundary values to set the intervals of the concavity.

This follows the first derivative test wherein:

Critical points occurs at x=a when f'(a) =0.

The graph for f(x) =` x^3(x-2)^4 ` is:

From the graph of f(x), the intervals of concavity are:

Concave up : (0, 0.8) and (2, +`oo` )

Concave down: (- `oo` , 0) and (0.8,2)

The critical points used as boundary values are:

inflection point at x=0

maximum point at x=0.8

minimum point at x=2

The graph of f"(x) = `6·x·(x - 2)^2·(7·x^2 - 12·x + 4)` or

`f(x) = 42x^5 - 240x^4 + 480x^3 - 384x^2 + 96x`

Using the graph of the second derivative to find the intervals of concavity, we follow:

f"(x) > 0 or graph of f"(x) below the x-axis for concave up

f"(x)< 0 graph of f"(x) above the x-axis for concave down

f"(x) =0 possible on inflection point.

Real inflection point occurs at x=a when there is change in concavity before and after x=a.

As shown in the graph, possible inflection points occurs at

x-values: 0, 0.45, 1.25, and 2 where the graph f"(x) intersects the x-axis.

Intervals of concavity:

Concave up: (0,0.45), (1.25 , +`oo` )

Concave down:(-`oo` , 0), (0.45 , 1.25).

Based on this, there are a real inflection points at x=0, 0.45, 1.26.