When is there one solution? - GRE Mathematics Subject Test

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[@slytherinbrian]

I did this in a slightly different (worse?) way. I solved for x using the Lambert W function and got x = e^(-W(-4 c)/4). The W function isn't defined for arguments < -1/e. It is equal to -1 at -1/e, has two values between -1/e and 0, and has a single value when the argument is greater than zero. Since all of the answer options are positive, the argument to W will be negative - and so it must be -1/e, the only negative argument that yields a single real value, and therefore c must be 1/(4e).

[@bogydan4223]

Actually the LambertW function has 0 real solutions on the interval (-infinity; -1/e), 2 solutions on the interval [-1/e; 0) and has one solution on the interval [0; infinity) and the argument of the function is -4c, therefore, c is either a number form (-infinity; -1/e) or the argument has to be -1/e, we still get one solutions for this value, so we set -4c=-1/e and we get c=1/4e

[@prof_ymca]

What a nice problem! Really puts into question if the examinee understands the meaning of a derivative!

[@camronmccutcheon2842]

Derivative if log(x) is not 1/x. Graphing tech confirms that the given solution does not work for this question. If you meant ln(x) = cx^4, then it does work. The way it is wtitten, the correct answer is c = log(e)/(4e)

[@mathoutloud]

Last time I checked that’s what it is for positive values of x.

[@anthonyflanders1347]

Many times in math log is used to denote the natural logarithm or base e

[@madhav-lp7un]

You should specify whether it is natural log or not​@@mathoutloud

[@mathoutloud]

It’s obvious from context here that it’s the natural log. It’s pretty much always the natural log when doing something purely mathematical, unless stated otherwise.

[@justabunga1]

The answer is correct, but somewhere in the middle of the step is incorrect because the derivative of log(x) is 1/(xln(10)). From there, if you differentiate both sides of the equation, you get 1/(xln(10))=4cx^3, so 4cx^4ln(10)=1. Now substitute cx^4=log(x), which is 4log(x)ln(10)=1. Note that log(x)=ln(x)/ln(10). Basically, 4ln(x)=1, so x=e^(1/4). If you raise both sides by 4, we have x^4=e. Putting back in the equation that was differentiated, we 4ce=1. Solving for c gives us c=1/(4e). Hence, the answer must be choice A.

[@anthonyflanders1347]

Here log is representing the natural logarithm and not the base 10 log very common in upper level mathematics

[@shivanshnigam4015]

Please mention the base of the log in the question

[@mathoutloud]

It is always e.

[@prathamkalgutkar7538]

@@mathoutloud That makes sense because I solved it using base of 10 cause you used log and not ln

[@alonewanderer4697]

@@prathamkalgutkar7538 yeah, depends on the field that you mainly studied in
maths often take base e for log, while physics often takes base 10

[@ramunasstulga8264]

​@@alonewanderer4697bro log is for log base 10, ln is for log base e

[@ramunasstulga8264]

​@@mathoutloudlog x usually means to the natural logarithm log x could also be to binary logarithm or base ten logarithm. Definitely not always and should be specified 😊

[@makakh2620]

Differentiating does not always work

[@asparkdeity8717]

But in this case, simple sketches show u that it does

[@GursimarSinghMiglani-ym7nu]

ez pz