Power Functions - Find Where they Equate on Interval

Algebra Level 3

$\begin{cases} f(x) = x^a + x^b + x^c + x^d \\ g(x) = a^x + b^x + c^x + d^x \end{cases}$

Functions $f(x)$ and $g(x)$ are defined as above, where $a=\frac 13$ , $b=\frac 12$ , $c=\frac 14$ , and $d=2$ , are continuous on $x \in [0,10]$ and $f(x)=g(x)$ at two points $x_1$ and $x_2$ . Find $x_1+x_2$ to the nearest $1/100$ th.

The answer is 5.54.

3 solutions

Steven Chase
Apr 25, 2020

Here's a hand-coded Newton Raphson algorithm. The function I call $f$ is the difference between the two functions given in the problem.

import math
import random

a = 1.0/3.0
b = 1.0/2.0
c = 1.0/4.0
d = 2.0

#########################################################################################################

x = 10.0*random.random()  # guess an initial value for x

for j in range(0,100):  # Newton-Raphson

    f = x**a + x**b + x**c + x**d - a**x - b**x - c**x - d**x   # function f = 0

    fp = a*(x**(a-1.0)) + b*(x**(b-1.0)) + c*(x**(c-1.0)) + d*(x**(d-1.0))   # derivative
    fp = fp - (a**x)*math.log(a) - (b**x)*math.log(b) - (c**x)*math.log(c) - (d**x)*math.log(d)

    x = x - f/fp  # update guess for x


f = x**a + x**b + x**c + x**d - a**x - b**x - c**x - d**x

print f
print x

#########################################################################################################


#>>> 
#-8.881784197e-16
#0.684878755615
#>>> ================================ RESTART ================================
#>>> 
#-3.5527136788e-15
#4.85210468951
#>>>

A Former Brilliant Member
Apr 25, 2020

Chew-Seong Cheong
Apr 25, 2020

Using Newton-Raphson method $x_{n+1} = x_n - \dfrac {f(x)}{f'(x)}$ with an Excel spreadsheet. $x_1 \approx 0.684878756$ , $x_2 \approx 4.85210469$ , and $x_1+x_2 \approx 5.536983445 \approx \boxed{5.54}$ .

Power Functions - Find Where they Equate on Interval

The answer is 5.54.

3 solutions

0 pending reports