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Please help! How to solve this function? show working please.?

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Consider the function f(x) =x^3−3x^2 + 4.
(i) Find f′(x) and f′′(x).
(ii) Find and classify all critical points of f(x).
(iii) Find any points of inflection.
(iv) Make a rough sketch of y=f(x) indicating all critical points and points of inflection.

asked May 22, 2014 in PRECALCULUS by anonymous

2 Answers

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You know how to differentiate? f(x) = x^n => f'(x) = n x^*n - 1). So here, (i) f'(x) = 3x² - 6x and f''(x) = 6x - 6. (ii) f'(x) = 0 => x = 0 or x = 2. x = 0 => f''(x) < 0 => this is a maximum point x = 2 => 6x - 6 > 0 => this is a minimum point. From the x's you can work out the y's and thus the full coordinates of these two critical points. (iii) Meanwhile, f''(x) = 0 => x = 1, so here is a point of inflection (again, work out the y value here). (iv) So, mark these three points on the graph and sketch the positive cubic that goes through them (like a wavy N shape with infinitely extending arms).
answered May 22, 2014 by anonymous
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The function is f ( x) = x3 - 3x2 + 4.

(i).

f ' (x) = 3x2 - 6x.

f '' (x) = 6x - 6.

(ii).

To find the critical numbers of f (x), Set  f ' (x) = 0, then find roots for the set of the equations.

f ' (x) = 3x2 - 6x = 0

3x(x - 2) = 0

Apply zero product property.

3x = 0 and x - 2 = 0

⇒ x = 0 and x = 2.

If x = 0, then y = f(0) = 03 - 3(0)2 + 4 = 0 - 0 + 4 = 4.

If x = 2, then y = f(2) = 23 - 3(2)2 + 4 = 8 - 12 + 4 = 0.

So, the critical points are (0, 4) and (2, 0).

(iii).

To find inflection of points, set f '' (x) = 0.

f '' (x) = 6x - 6 = 0

6x = 6

⇒ x = 6/6 = 1.

To find the y - coordinate of this point inflection, find f (1).

f ( 1) = 13 - 3(1)2 + 4 = 1 - 3 + 4 = 5 - 3 = 2..

So, the point(s) of inflection is (1, 2).

(iv).

The function is f ( x) = y = x3 - 3x2 + 4.

Make the table, Choose different values of x and obtain random y - values.

x

y = x3 - 3x2 + 4

(x, y)

- 1.1

y = (-1.1)3 - 3(- 1.1)2 + 4 = - 0.961

(- 1.1, - 0.961)

- 1

y = (-1)3 - 3(- 1)2 + 4 = -1 - 3 + 4 = 4 - 4 = 0

(- 1, 0)

0

y = (0)3 - 3(0)2 + 4 = 4

(0, 4)

1

y = (1)3 - 3(1)2 + 4 = 2

(1, 2)

2

y = (2)3 - 3(2)2 + 4 = 0

(2, 0)

3

y = (3)3 - 3(3)2 + 4 = 4

(3, 4)

  • Draw a coordinate plane.
  • Plot the points
  • And connected these points with a smooth curve.

Graph:

graph the function y=xsqrt(9-x^2)

answered May 22, 2014 by lilly Expert

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