F ( x ) = 8 x tan x , − π / 2 < x < π / 2 (a) Find the vertical asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = Find the horizontal asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) y = (b) Find the interval where the function is increasing. (Enter your answer using interval notation.) Find the interval where the function is decreasing. (Enter your answer using interval notation.) (c) Find the local maximum and minimum values. (If an answer does not exist, enter DNE.) local maximum value local minimum value (d) Find the interval where the function is concave up. (Enter your answer using interval notation.)

a) The vertical asymptote(s) are x = − π / 2   and  x = π / 2

b) The interval where the function is decreasing is (- π / 2, 0)

The interval where the function is increasing is (0, π / 2)

c) Minimum: P (0, 0)

d) The function is concave up in the the interval (- π / 2, π / 2)

Step-by-step explanation:

a) The vertical asymptote(s) are

 lim (x→− π/2) (8x*tan(x)) = 8*(− π/2)*tan(− π/2) = -4 π*(- ∞) → - ∞

x = − π / 2

lim (x→ π/2) (8x*tan(x)) = 8*(π/2)*tan( π/2) = 4 π*(∞) → + ∞

x = π / 2

The horizontal asymptote(s) does not exist (DNE).

b) f'(x) = (8x*tan(x))' = Sin(x) + x*Cos(x)

If x = - π / 4   where − π / 2 < - π / 4 < 0  we have

f'(- π / 4) = Sin(- π / 4) + (- π / 4)*Cos(- π / 4)

f'(- π / 4) = -(√2/2) - (π / 4)(√2/2) = -(√2/2)((4 + π) / 4) < 0

The interval where the function is decreasing is (- π / 2, 0)

If x =  π / 4   where  0 < π / 4 < π / 2  we have

f'( π / 4) = Sin( π / 4) + ( π / 4)*Cos( π / 4)

f'(- π / 4) = (√2/2) + (π / 4)(√2/2) = (√2/2)((4 + π) / 4) > 0

The interval where the function is increasing is (0, π / 2)

c) f(x) = 8x*tan(x)

f'(x) = (8x*tan(x))' = 0

⇒ tan(x) + x*Sec²(x) = 0

⇒ (Sin(x)/Cos(x)) + (x/Cos²(x)) = 0

⇒ Sin(x) + x*Cos(x) = 0   ⇒  x = 0

f(0) = 8*0*tan(0) = 0

Minimum: P (0, 0)

d) f''(x) = (Sin(x) + x*Cos(x))'

⇒ f''(x) = Cos(x) + Cos(x) - x*Sin(x)

⇒ f''(x) = 2*Cos(x) - x*Sin(x)

If x = 0 where − π / 2 < 0 < π / 2  we have

f''(0) = 2*Cos(0) - 0*Sin(0) = 2*(1) - 0 = 2 > 0

Then, the function is concave up in the the interval (- π / 2, π / 2)

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