Let f, g and h be real-valued functions defined on the interval [0, 1] by f \left(\right. x \left.\right) = e^{x^{2}} + e^{- x^{2}} , g \left(\right. x \left.\right) = x \textrm{ } e^{x^{2}} + e^{- x^{2}} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } h \left(\right. x \left.\right) = 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

Engineering

Mathematics

Maxima and Minima

Question

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

A

a = b = c

B

a = b and c $\neq$ b

C

a $\neq$ b and c $\neq$ b

D

a = c and a $\neq$ b

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$f (x) = e^{x^{2}} + e^{- x^{2}}$

$f' (x) = 2 {xe}^{x^{2}} - 2 {xe}^{- x^{2}}$

$f' (x) = 2 x (e^{x^{2}} - e^{- x^{2}})$

$f' (x) > 0 \forall x \in [0, 1]$

$so, f (x)]_{\max} = e + \frac{1}{e} = a$

$g (x) = x e^{x^{2}} + e^{- x^{2}}$

$g' (x) = 2 x^{2} e^{x^{2}} + e^{x^{2}} - 2 {xe}^{- x^{2}}$

$g' (x) > 0 \forall x \in [0, 1]$

$g (x)]_{\max} = e + \frac{1}{e} = b$

$h (x) = x^{2} e^{x^{2}} + e^{- x^{2}}$

$h' (x) = 2 x^{3} e^{x^{2}} + 2 x e^{x^{2}} - 2 x e^{- x^{2}}$

$h' (x) > 0 \forall x \in [0, 1]$

h (x)]_max = h(1) = e + = c

so, a = b = c Ans.

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