Testing your intelligence

1. 17/9
2. 4/9
3. 1/12
4. 1/4

the first one simlifies to =

3∫f(x)dx

= 3(₀∫^1/3(x-1)²dx+_1/3∫^2/32x(1-x)dx+_2/3∫¹x²dx

some mistake, limits misplaced i think,

yes i have corrected it now

for (2)

min value of f(x) comes out to be 4/9
max value of g(x) comes out to be 1/4

for (3)

it is nothing but

₀∫¹g(x)dx

which is equal to

₀∫^1/2x²dx+_1/2∫¹(x-1)²dx

i think thre is some thing wrong with your picture rohan.

for part 4
from the graph we can see that f(x)
lies above g(x) in both (-1,0) and (0,1)

and we can figure out that if the point of non differentiability of g(x) is made to go 1/4 units upward
it will cut f(x) at exactly one point