Question

Let, g(x) = 2f(x/2) + f(2-x) and f"(x)< 0 for all x € (0,2) .

Find the interval of increase and decrease of g(x).


❌❌NO SPAMMING❌❌​

Answer 1

See the reference.........

Answer 2

Answer:

Let, g(x) = 2f(x/2) + f(2-x) and f"(x)< 0 for all x € (0,2) .

Step-by-step explanation:

since  f"(x)< 0 for all x € (0,2) .

         f'[x] is decreasing function

g'(x) = 2f'(x/2)[1/2] + f'(2-x)[-1]

         = f'(x/2) - f'(2-x)

since f'[x] is decreasing function  when x/2 >2-x       f'(x/2) < f'(2-x)

that is  x>4-2x  

           3x>4

            x>4/3         f'(x/2) - f'(2-x) < 0    g'(x) < 0  

  so [4/3 ,2 ]   g(x) is decreasing since

when f'[x] is decreasing function  when x/2 <2-x       f'(x/2) > f'(2-x)

that is  x<4-2x  

           3x<4

            x<4/3         f'(x/2) - f'(2-x) > 0    g'(x) >0  

  so [0<4/3  ]   g(x) is   increasing