Boomerang

Calculus Level 3

Solve for x > 0 x>0 and answer up to four decimal places.

x = x 5 + x 10 + x 100 \large x=\sqrt [ 5 ]{ x } +\sqrt [ 10 ]{ x } +\sqrt [ 100 ]{ x }


The answer is 3.4222.

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Department 8 by Department 8 , 27, Israel

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1 solution

David Holcer
Jun 15, 2018

The following (python3) code automates the Newton method for 15000 trials to give an accurate value of x. 

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def f(x):
    return x**(1/5)+x**(1/10)+x**(1/100)-x
def f_prime(x):
    return (1/5)*x**(-4/5)+ (1/10)*x**(-9/10) + (1/100)*x**(-99/100) -1

x_n=1
i=0
lim=15000
while i<lim:
    x_n=x_n-(f(x_n)/f_prime(x_n))
    i+=1
print(x_n)

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