Question

f(x) = ((ln(7x−x^2)/ 12))^3/2 is it an increasing function?

Answer 1

given  f(x) = (ln((7x - x²)/12)) ^3/2

 f'(x) =  3/2[ ln (7x - x²)/12]^1/2 {1/7x - x²} (7- 2x) = 0  

now g(x) = 7x - x²/12 is  increasing function  for x<7/2

g(x) = 1 ⇒   x =  0.0145  ⇒  g(x) < 1 for x < 0.0145

             that implies ln((7x-x²)/12)  is negative for x  < 0.0145

              and positive for x > 0.0145

now  case 1:  x < 0.0145

               f'(x) <  0

             - (7 - 2x)/ 7x -x² > 0  ⇒  (2x-7)/ x ( x-7) < 0

               x ∈  (0 , 7/2) U ( 7, ∞)

but that do not matches with the condition so I think it gonna be wrong

           now f(x) > 0  for x ∈ (-∞ , 0) U (7 , ∞)

intersection with condition gives x ∈ (-∞ , 0 ) 
therefore f(x) is increasing for x < 0

now  for case : 2  x > 0.0145

           f'(x) > 0 ⇒ (7-2x) / 7x - x² > 0 ⇒  (2x-7)/ x (x-7) > 0

                     that gives x ∈ ( 0 , 7/2)  U (7 , ∞ )

so f(x) is increasing for x  ∈ (0, 7/2) U (7, ∞)

and f'(x) < 0  for x∈  (-∞ ,.0 ) U (7/2 , 7 )

intersection with the condition gives x ∈ (7/2 , 7 )

so f(x) is decreasing for x ∈ (7/2 , 7 )

hence in conclusion  f(x) is decreasing for x  ∈  (7/2 ,7)

and increasing for x ∈  (-∞ , 0) U ( 0 , 7/2) U (7, ∞)


and its neither strictly incresing nor strictly decreasing ..........


hope ma'am u will mark it as brainliest.........if queries please ask ma'am..