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)

arrange the equations with like terms in columns.

analyze the coefficients of x or y.

add the equations and solve for the remaining variable.

substitute the value into either equation and solve.

check the solution.

step-by-step explanation:

30005

yes ok