The minimum value of 4 e^ 2x + 9 e^ - 2x is - Vidyadip

Question

The minimum value of $4{e^{2x}} + 9{e^{ - 2x}}$ is

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Answer

b
(b) Let $f(x) = 4{e^{2x}} + 9{e^{ - 2x}}$

$\therefore $ $f'(x) = 8{e^{2x}} - 18{e^{ - 2x}}$

Put $f'(x) = 0 \Rightarrow 8{e^{2x}} - 18{e^{ - 2x}} = 0$

${e^{2x}} = 3/2 \Rightarrow x = \log {(3/2)^{1/2}}$

Again $f''(x) = 16{e^{2x}} + 36{e^{ - 2x}} > 0$

Now $f(\log {(3/2)^{1/2}}) = 4{e^{2.(\log {{(3/2)}^{1/2}})}} + 9{e^{ - 2(\log {{(3/2)}^{1/2}})}}$

$= 4 \times \frac{3}{2} + 9 \times \frac{2}{3} = 6 + 6 = 12$

Hence minimum value $= 12.$

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