The maximum value M of 3^x+5^x-9^x+15^x-25^x, | vedclass

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Question

(b)

Let

$f(x)=3^x+5^x-9^x+15^x-25^{-x}$

$f(x)=3^x+5^x-3^{2 x}+3^x \cdot 5^x-5^{2 x}$

Maximum value of $f(x)$ is $M$.

$	herefore \quad

Vedclass · Accepted Answer

$0 < M < 2$

Vedclass · Answer

(b)

Let

$f(x)=3^x+5^x-9^x+15^x-25^{-x}$

$f(x)=3^x+5^x-3^{2 x}+3^x \cdot 5^x-5^{2 x}$

Maximum value of $f(x)$ is $M$.

$	herefore \quad M=3^x+5^x-3^{2 x}-3^x \cdot 5^x-5^{2 x}$

$M=a+b-a^2+a b-b^2$

$\left[\because 3^x=a, 5^x=bight]$

$\Rightarrow \quad a+b+a b-\left(a^2+b^2ight)$

$\Rightarrow \quad M \leq a+b+a b-2 a b$

$\left.M \leq a+b-a b-\left(a^2+b^2ight) \leq-2 a bight]$

$\Rightarrow \quad M \leq 1-(a-1)(b-1)$

So, maximum value of $M$ is 1 at $x=0$.

$	herefore \quad 0 < M < 2$

The maximum value $M$ of $3^x+5^x-9^x+15^x-25^x$, as $x$ varies over reals, satisfies

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