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What is the maximum value of

$(ax+b)^\frac12 + (cx+d)^\frac12$

I did it using the concept $AM>=GM$

$\frac{(ax+b)^\frac12+(cx+d)^\frac12}2 \geq ((ax+b)^\frac12(cx+d)^\frac12)^\frac12$

And on solving this I found that $x\leq(b-d)/(c-a)$

And when $x$ is maximum value of the expression will be maximum

But this was not the answer. Somebody help me, please.

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  • $\begingroup$ Did you try with algebra ? $\endgroup$
    – Claude Leibovici
    Commented Aug 9, 2019 at 9:51
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    $\begingroup$ Do you mean the minimum value when $x \in \mathbb R$? Or is there some restriction on $a,b,c,d$? $\endgroup$
    – Toby Mak
    Commented Aug 9, 2019 at 9:54
  • $\begingroup$ Consider the function $$f(x)=\sqrt{ax+b}+\sqrt{cx+d}$$ and differentiate with respect to $x$ $\endgroup$
    – Dr. Sonnhard Graubner
    Commented Aug 9, 2019 at 9:57
  • $\begingroup$ Differentiation gives minima $\endgroup$
    – Kshitij Singh
    Commented Aug 9, 2019 at 10:10
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    $\begingroup$ if $a,b>0$ then $ (ax+b)^\frac12 + (cx+d)^\frac12$ is not bounded and increasing so it does not make sense to look for a maximum value. $\endgroup$
    – miracle173
    Commented Aug 9, 2019 at 12:17

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