<s>[INST] How to find real roots? $4 ^{x} +6 ^{x ^{2}} =5 ^{x} +5 ^{x ^{2}}$ How to find real roots? 
$$4 ^{x} +6 ^{x ^{2}} =5 ^{x} +5 ^{x ^{2}}$$
I try LMVT but vary difficult
 [/INST] The following argument is admittedly on the intuitive side, using concavity mixed with "flatter than" comparisons.
If $x<0$ then since $x^2>0$ we have
$$5^x-4^x<0<6^{x^2}-5^{x^2}.$$
If $0<x<1$ then in fact $0<x^2<x<1$, and the functions $u^x$ and $u^{x^2}$ are both concave down, with $u^{x^2}$ being flatter between 5 and 6 than is $u^x$ there. So
$$6^{x^2}-5^{x^2}<6^x-5^x<5^x-4^x.$$
Finally if $x>1$ then in fact $1<x<x^2$ and the functions $u^x$ and $u^{x^2}$ are both concave up, with $u^x$ being flatter between 5 and 6 than is $u^{x^2}$ there. So
$$6^{x^2}-5^{x^2}>6^x-5^x>5^x-4^x.$$
If the above (admittedly vague) arguments are right, we have ruled out all possibilities other than $x=0,1$ for the equation $5^x-4^x=6^{x^2}-5^{x^2}$.
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