Function approximating camels humps?












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I want to construct a function with two maxima and a minima in between that would approximate the two humps of a camel.
In addition, I would like that there would be some effort (not a polynomial) to find the derivatives. And I would also like that the stationary points and inflection points can be evaluated without a calculator.



The closest I got was the function $$ln(8 - frac{1}{4}x^4 + 3 x^2 - 4 x),$$
but the inflection points are not nice.






functions graphing-functions stationary-point







share|cite|improve this question









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  • $begingroup$
    I think tasks like this are closer to art than mathematics. It's an artform, choosing the correct functions and coefficients.
    $endgroup$
    – Matti P.
    Jan 30 at 9:15










  • $begingroup$
    Well, I'm not trying to get the real-life curve, I just want to create an exercise.
    $endgroup$
    – Jake B.
    Jan 30 at 14:07
















-2












$begingroup$


I want to construct a function with two maxima and a minima in between that would approximate the two humps of a camel.
In addition, I would like that there would be some effort (not a polynomial) to find the derivatives. And I would also like that the stationary points and inflection points can be evaluated without a calculator.



The closest I got was the function $$ln(8 - frac{1}{4}x^4 + 3 x^2 - 4 x),$$
but the inflection points are not nice.






functions graphing-functions stationary-point







share|cite|improve this question









$endgroup$












  • $begingroup$
    I think tasks like this are closer to art than mathematics. It's an artform, choosing the correct functions and coefficients.
    $endgroup$
    – Matti P.
    Jan 30 at 9:15










  • $begingroup$
    Well, I'm not trying to get the real-life curve, I just want to create an exercise.
    $endgroup$
    – Jake B.
    Jan 30 at 14:07














-2












-2








-2





$begingroup$


I want to construct a function with two maxima and a minima in between that would approximate the two humps of a camel.
In addition, I would like that there would be some effort (not a polynomial) to find the derivatives. And I would also like that the stationary points and inflection points can be evaluated without a calculator.



The closest I got was the function $$ln(8 - frac{1}{4}x^4 + 3 x^2 - 4 x),$$
but the inflection points are not nice.






functions graphing-functions stationary-point







share|cite|improve this question









$endgroup$




I want to construct a function with two maxima and a minima in between that would approximate the two humps of a camel.
In addition, I would like that there would be some effort (not a polynomial) to find the derivatives. And I would also like that the stationary points and inflection points can be evaluated without a calculator.



The closest I got was the function $$ln(8 - frac{1}{4}x^4 + 3 x^2 - 4 x),$$
but the inflection points are not nice.






functions graphing-functions stationary-point




functions graphing-functions stationary-point






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 30 at 9:13









Jake B.Jake B.

1786




1786












  • $begingroup$
    I think tasks like this are closer to art than mathematics. It's an artform, choosing the correct functions and coefficients.
    $endgroup$
    – Matti P.
    Jan 30 at 9:15










  • $begingroup$
    Well, I'm not trying to get the real-life curve, I just want to create an exercise.
    $endgroup$
    – Jake B.
    Jan 30 at 14:07


















  • $begingroup$
    I think tasks like this are closer to art than mathematics. It's an artform, choosing the correct functions and coefficients.
    $endgroup$
    – Matti P.
    Jan 30 at 9:15










  • $begingroup$
    Well, I'm not trying to get the real-life curve, I just want to create an exercise.
    $endgroup$
    – Jake B.
    Jan 30 at 14:07
















$begingroup$
I think tasks like this are closer to art than mathematics. It's an artform, choosing the correct functions and coefficients.
$endgroup$
– Matti P.
Jan 30 at 9:15




$begingroup$
I think tasks like this are closer to art than mathematics. It's an artform, choosing the correct functions and coefficients.
$endgroup$
– Matti P.
Jan 30 at 9:15












$begingroup$
Well, I'm not trying to get the real-life curve, I just want to create an exercise.
$endgroup$
– Jake B.
Jan 30 at 14:07




$begingroup$
Well, I'm not trying to get the real-life curve, I just want to create an exercise.
$endgroup$
– Jake B.
Jan 30 at 14:07










1 Answer
1






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oldest

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1












$begingroup$

How about this function? $$f(x)=2e^{-x^2}+2e^{-(x-2)^2}-e^{-(x-1)^2}$$



Its graph looks like this:



enter image description here






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Good idea to use normal distributions!
    $endgroup$
    – Jake B.
    Feb 24 at 17:51












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

How about this function? $$f(x)=2e^{-x^2}+2e^{-(x-2)^2}-e^{-(x-1)^2}$$



Its graph looks like this:



enter image description here






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Good idea to use normal distributions!
    $endgroup$
    – Jake B.
    Feb 24 at 17:51
















1












$begingroup$

How about this function? $$f(x)=2e^{-x^2}+2e^{-(x-2)^2}-e^{-(x-1)^2}$$



Its graph looks like this:



enter image description here






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Good idea to use normal distributions!
    $endgroup$
    – Jake B.
    Feb 24 at 17:51














1












1








1





$begingroup$

How about this function? $$f(x)=2e^{-x^2}+2e^{-(x-2)^2}-e^{-(x-1)^2}$$



Its graph looks like this:



enter image description here






share|cite|improve this answer









$endgroup$



How about this function? $$f(x)=2e^{-x^2}+2e^{-(x-2)^2}-e^{-(x-1)^2}$$



Its graph looks like this:



enter image description here







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Feb 22 at 23:05









Robert HowardRobert Howard

2,2933935




2,2933935








  • 1




    $begingroup$
    Good idea to use normal distributions!
    $endgroup$
    – Jake B.
    Feb 24 at 17:51














  • 1




    $begingroup$
    Good idea to use normal distributions!
    $endgroup$
    – Jake B.
    Feb 24 at 17:51








1




1




$begingroup$
Good idea to use normal distributions!
$endgroup$
– Jake B.
Feb 24 at 17:51




$begingroup$
Good idea to use normal distributions!
$endgroup$
– Jake B.
Feb 24 at 17:51


















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