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Question Number 143208 by Gbenga last updated on 11/Jun/21

3^x +4x−3=x^4 find x

$$\mathrm{3}^{{x}} +\mathrm{4}{x}−\mathrm{3}={x}^{\mathrm{4}} \\ $$$${find}\:{x} \\ $$

Answered by MJS_new last updated on 11/Jun/21

you can only approximate 3^x =x^4 −4x+3 f(x)=3^x is strictly increasing ∀x∈R g(x)=x^4 −4x+3 has an absolute minimum at ((1),(0) ) and is strictly decreasing for x<1 and strictly increasing for x>1 f(0)=1∧g(0)=3 ⇒ we have to search for x>0 f(8)=6561∧g(8)=4067 ⇒ x<8 I get x_1 ≈.376830 x_2 ≈1.87395 x_3 ≈7.15659

$$\mathrm{you}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate} \\ $$$$\mathrm{3}^{{x}} ={x}^{\mathrm{4}} −\mathrm{4}{x}+\mathrm{3} \\ $$$${f}\left({x}\right)=\mathrm{3}^{{x}} \:\mathrm{is}\:\mathrm{strictly}\:\mathrm{increasing}\:\forall{x}\in\mathbb{R} \\ $$$${g}\left({x}\right)={x}^{\mathrm{4}} −\mathrm{4}{x}+\mathrm{3}\:\mathrm{has}\:\mathrm{an}\:\mathrm{absolute}\:\mathrm{minimum} \\ $$$$\mathrm{at}\:\begin{pmatrix}{\mathrm{1}}\\{\mathrm{0}}\end{pmatrix}\:\:\mathrm{and}\:\mathrm{is}\:\mathrm{strictly}\:\mathrm{decreasing}\:\mathrm{for}\:{x}<\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{strictly}\:\mathrm{increasing}\:\mathrm{for}\:{x}>\mathrm{1} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{1}\wedge{g}\left(\mathrm{0}\right)=\mathrm{3}\:\Rightarrow\:\mathrm{we}\:\mathrm{have}\:\mathrm{to}\:\mathrm{search}\:\mathrm{for}\:{x}>\mathrm{0} \\ $$$${f}\left(\mathrm{8}\right)=\mathrm{6561}\wedge{g}\left(\mathrm{8}\right)=\mathrm{4067}\:\Rightarrow\:{x}<\mathrm{8} \\ $$$$\mathrm{I}\:\mathrm{get} \\ $$$${x}_{\mathrm{1}} \approx.\mathrm{376830} \\ $$$${x}_{\mathrm{2}} \approx\mathrm{1}.\mathrm{87395} \\ $$$${x}_{\mathrm{3}} \approx\mathrm{7}.\mathrm{15659} \\ $$

Commented by Gbenga last updated on 11/Jun/21

thanks

$${thanks} \\ $$

Commented by Gbenga last updated on 11/Jun/21

find x in x+e^x +x^4 −4=0

$${find}\:{x}\:{in}\:{x}+{e}^{{x}} +{x}^{\mathrm{4}} −\mathrm{4}=\mathrm{0} \\ $$

Commented by mr W last updated on 11/Jun/21

there is no standard exact way for solution! therefore more such questions make no sense. you may ask as next find x in x+e^(−x) +x^4 −4=0 find x in x+sin x+x^4 −4=0 find x in x+cosh x+x^4 −4=0 ......

$${there}\:{is}\:{no}\:{standard}\:{exact}\:{way}\:{for} \\ $$$${solution}!\:{therefore}\:{more}\:{such}\: \\ $$$${questions}\:{make}\:{no}\:{sense}.\: \\ $$$${you}\:{may}\:{ask}\:{as}\:{next} \\ $$$${find}\:{x}\:{in}\:{x}+{e}^{−{x}} +{x}^{\mathrm{4}} −\mathrm{4}=\mathrm{0} \\ $$$${find}\:{x}\:{in}\:{x}+\mathrm{sin}\:{x}+{x}^{\mathrm{4}} −\mathrm{4}=\mathrm{0} \\ $$$${find}\:{x}\:{in}\:{x}+\mathrm{cosh}\:{x}+{x}^{\mathrm{4}} −\mathrm{4}=\mathrm{0} \\ $$$$...... \\ $$

Commented by Gbenga last updated on 12/Jun/21

0k sir

$$\mathrm{0}{k}\:{sir}\: \\ $$

Commented by Gbenga last updated on 12/Jun/21

but it was also sent to me by a friend

$${but}\:{it}\:{was}\:{also}\:{sent}\:{to}\:{me}\:{by}\:{a}\:{friend} \\ $$

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