Question

f(x) = 2x^3 + 3x^2 - 12x
(a) Find the intervals on which f is increasing.
( , ) ( , )

(b) Find the interval on which f is decreasing.
( , )

(c) Find the local minimum and maximum values of f.
(min)
20 (max)

(d) Find the inflection point.
( , )

(e) Find the interval on which f is concave up.
( , )

(f) Find the interval on which f is concave down.

Answer 1

Is this correct
f(x) = 2x^3 + 3x^2 - 12x

OK, here goes.
a) To answer this question you need to first find the slope of the line, and then the section of the line with a positive slope.

The slope is the derivative of the equation f(x)
f'(x) = 6x^2 + 6x - 12

Now find where f'(x) is positive, or f'(x) > 0. Do this by setting
6x^2 + 6x - 12 > 0 and solve for x. You can do the work.

b) This problem is similar to a) except that you need to find where the slope is negative.

f'(x) < 0
or,
6x^2 + 6x - 12 < 0

Solve for x

c) To find the locations of maxima or minima, you need to find the point(s) where the slope is zero.

so, set f'(x) = 0
6x^2 + 6x - 12 = 0

Now solve for x. Since this is a quadratic, there will be two roots (values of x). These are the locations of the maxima and minima.

d) The inflection point is the location of maximum slope.

slope = f'(x) = 6x^2 + 6x - 12

As usual, to find a maximum take the derivative of the equation and set equal to zero.

f''(x) = derivative of f'(x) = 12x + 6
12x + 6 = 0
x = -0.5 This is the location of the point of inflection

e) We found the maxima and minima points in step c. To find out if the point is a maximum or a minimum, have to insert that value into f''(x). If the result is negative, the value is a maxima. If the value is positive, the result is a minima.

So, insert each of the two values calculated in step c into

f''(x) = 12x + 6

You will get two answers, one for each value. The negative value is the location of the maxima, the positive is the location of the minima. The interval over which the line is concave up includes the minima up to the point of inflection. The interval over which the line is concave down includes the maxima up to the point of inflection.

Answer 2

I have also posted it in yahoo answer but think... plzz suggest

Answer 3

i think u r right

Answer 4

thankx

Answer 5

if yess u can give me medal and if no reasons