Application of Derivatives 4 Question 18
19. Let $f, g$ and $h$ be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^{2}}+e^{-x^{2}}, g(x)=x e^{x^{2}}+e^{-x^{2}}$ and $h(x)=x^{2} e^{x^{2}}+e^{-x^{2}}$. If $a, b$ and $c$ denote respectively, the absolute maximum of $f, g$ and $h$ on $[0,1]$, then
(2010)
(a) $a=b$ and $c \neq b$
(b) $a=c$ and $a \neq b$
(c) $a \neq b$ and $c \neq b$
(d) $a=b=c$
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Answer:
Correct Answer: 19. (a)
Solution:
- Given function, $f(x)=e^{x^{2}}+e^{-x^{2}}, g(x)=x e^{x^{2}}+e^{-x^{2}}$ and $h(x)=x^{2} e^{x^{2}}+e^{-x^{2}}$ are strictly increasing on $[0,1]$.
Hence, at $x=1$, the given function attains absolute maximum all equal to $e+1 / e$.
$$ \Rightarrow \quad a=b=c $$
