KVPY Sample Paper KVPY Stream-SX Model Paper-31

Let \[f''(x)>0\] and \[\phi (x)=f(x)+f(2-x),x\in (0,2)\] be a function, then the function \[\phi (x)\] is:

A) increasing in (0, 1) and decreasing (1,2)

B) decreasing in (0, 1) and increasing (1,2)

C) increasing in (0, 2)

D) decreasing in (0, 2)

Correct Answer: B

Solution :

\[f''(x)>0,y=f(x);x\in (0,2)\]

\[\phi (x)=f(x)+f(2-x)\]

\[\phi '(x)=f'(x)-f'(2-x)\]

for\[\phi (x)\]to be increasing

\[\phi '(x)>0\]\[\Rightarrow \]\[f'(x)>f'(2-x)\]\[\Rightarrow \]\[x>2-x\]

\[(f'(x)\]is increasing in (0, 2)\[\Rightarrow \]\[x>1\]\[\Rightarrow \]\[x\in (1,2)\]

For\[\phi '(x)\]to be decreasing

\[\phi '(x)<0\]\[\Rightarrow \]\[f'(x)<f'(2-x)\]

\[\therefore \] \[x\in (0,1).\]