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?
- There exists a function f∈S such that Xf=0
- For every function f∈S, we have Xf ≤ 2
- There exists a function f∈S such that Xf=2
- There does not exist any function f is S such that Xf=1
The Correct Option is A, B, C
Solution and Explanation
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).
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C