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

Question

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.

Answer 1

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).

Answer 2

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: