Question

Let S be the set of all twice differentiable functions f from R to R such that $\frac{d^2f}{dx^2}(x)>0$ for all x∈(-1,1). For f∈S, let Xf be the number of points x∈(-1,1) for which f(x)=x.Then which of the following statements is(are) true?

• A. There exists a function f∈S such that Xf=0
• B. For every function f∈S, we have Xf ≤ 2
• C. There exists a function f∈S such that Xf=2
• D. There does not exist any function f is S such that Xf=1

Answer 1

Answer: (A), (B), (C)

There exists a function f∈S such that Xf=0

Answer 2

The accurate choices are (A), (B), and (C). Given that f ′′(x)>0 and f(x)− x =0, we need to determine the number of solutions. Let g(x)=f(x)− x , which implies g ′(x)=f ′(x)−1 and g ′′(x)=f ′′(x)>0. This indicates the presence of concave possibilities

The correct options are (A),(B) and (C).