Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Continuity & Differentiability
Let H ∶ R → R be the function given by \(H(x)=\frac{1}{2}\left(e^{x}+e^{-x}\right)\) for x ∈ R. Let f ∶ R → R be defined by \(f(x)=\int_{0}^{\pi} H(x \sin \theta) d \theta\) for x ∈ R. Then which one of the following is true?
1
xf′′(x) + f′(x) + xf(x) = 0 for all x ∈ R.
2
xf′′(x) − f′(x) + xf(x) = 0 for all x ∈ R.
3
xf′′(x) + f′(x) − xf(x) = 0 for all x ∈ R.
4
xf′′(x) − f′(x) − xf(x) = 0 for all x ∈ R.