Find the intervals on which f is increasing and decreasing. superimpose the graphs of f and f prime to verify your work. f(x)= -4 - x + 3x^2

For the given function

-----------------------(1)

we have to see when it will increase and decrease.

For that we will first find derivative of function

Now derivative of first term in f(x) which is -4 will be 0 as its constant.

then derivative of x is 1, so derivative of -x will be -1

To derivate last term of function, , we will use power rule formula:

so

constant 3 will come as it is, so derivative of 3x^2 wil be 3(2x)= 6x.

So f'(x) = 0 -1 +6x

f'(x) = -1 + 6x------------------------------(2)

For function to be increasing f'(x) should be positive and for function to be decreasing f'(x) shoould be negative

So we will first find where f'(x) = 0

So put 0 in f'(x) place in equation (2)

0 = -1 + 6x

Now solve for x as shown

0 +1 = -1 + 6x +1

1 = 6x

So now we can have two regions as shown on either side of 1/6 on number line

_____________I_____________________II__________

So to test region I, pick any number to the left side of . For example lets take 0. now plug 0 inx place in f'(x) equation given by (2)

f'(x)= -1 + 6x

f'(0) = -1 + 6(0) = -1+0 = -1 which is negative. So since f'(x) we got as negative for region I, so this will be decreasing.

Interval for region I will be (-∞, )

----------------------------------------------------------------------------------------------------------

similarly now test region II. For that pick any number to the right of , lets take 1. So plug 1 in x place in f'(x) equation given by (2)

f'(x) = -1 + 6x

f'(1) = -1 +6(1)= -1+6 = 5 which is positive so we will have f(x) increasing in this region.

Interval for region II will be (, ∞)

------------------------------------------------------------------------

Graphs of and are shown in attachment. So you can clearly see in graph that f'(x) is negative (below x axis) from -∞ till so f(x) should be decreasing in this part which we can see from f(x) graph that its decreasing from -∞ till .

Similarly f'(x) is positive( above x axis) beyond so its increasing beyond which we can veryify from f(x) graph we can see that its increasing from till ∞. Hence verified

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

jwjwkwkwkkkdjebd b jdjejdndj he jenej