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 an

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 [2010]

Smart Achievers · Accepted Answer

(4)

$f (x) = e^{x^{2}} + e^{- x^{2}}$ $\Rightarrow f' (x) = 2 x (e^{x^{2}} - e^{- x^{2}}) \geq 0,$ $\forall x \in [0, 1]$

$∴$ $f (x) is an increasing function on [0, 1]$

$g$ $f_{\max} = f (1) = e + \frac{1}{e} = a;$ $g (x) = x e^{x^{2}} + e^{- x^{2}}$

$\Rightarrow g' (x) = (2 x^{2} + 1) e^{x^{2}} - 2 x e^{- x^{2}} \geq 0,$ $\forall x \in [0, 1]$

$∴$ $g (x) is an increasing function on [0, 1]$

$∴$ $g_{\max} = g (1) = e + \frac{1}{e} = b$

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

$\Rightarrow h' (x) = 2 x [e^{x^{2}} (1 + x^{2}) - e^{- x^{2}}] \geq 0,$ $\forall x \in [0, 1]$

$∴$ $h (x) is an increasing function on [0, 1]$

$∴$ $h_{\max} = h (1) = e + \frac{1}{e} = c$

$∴$ $a = b = c$

Solution

1 $a = b$ and $c \neq b$

2 $a = c$ and $a \neq b$

3 $a \neq b$ and $c \neq b$

4 $a = b = c$

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let f, g and h be real-valued functions defined on the interval [0, 1] by f(x)=ex2+e-x2, g(x)=xex2+e-x2 and h(x)=x2ex2+e-x2. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then [2010]

1 a=b and c≠b

2 a=c and a≠b

3 a≠b and c≠b

4 a=b=c

Want Us to Email you About Special Offers & Updates?

1 $a = b$ and $c \neq b$

2 $a = c$ and $a \neq b$

3 $a \neq b$ and $c \neq b$

4 $a = b = c$