Verify that x^3 satisfies the equation xf\' (x) = 3f(x). (Again, to verify a cla
ID: 2874867 • Letter: V
Question
Verify that x^3 satisfies the equation xf' (x) = 3f(x). (Again, to verify a claim, you should write down an explanation that would convince a classmate that disagrees with the claim that they’re wrong and you’re right.)
(b) Write down two other functions that satisfy the equation xf' (x) = 3f(x).
(c) For each of the functions you wrote down, what is f(0)?
(d) Write down two functions that satisfy the equation xf' (x) = 5f(x).
(e) Let k be a nonnegative whole number. Write down two functions that satisfy the equation xf' (x) = kf(x). Your answer should be in terms of k.
Explanation / Answer
a) f(x) = x3 ; then f' (x) = 3x2
x.f' (x) = x.3x2 = 3x3-----------------------------(i)
3f(x) = 3. x3 = 3x3-----------------------------(ii)
from (i) and (ii), we get
xf' (x) = 3f(x) ( when ever we find the derivate of any function which is a variable of x raised to the power 3, we get 3 times the function raised to the original power minus one (Since d/dx (xn) = nxn-1; So for every function with the variable of x raised to 3, xf' (x) = 3f(x).
b) we can take any constant mulitplied by the variable of x3 , this will satisfy the equation xf' (x) = 3f(x).
For eg. f(x) = 2x3 , f(x) = 3x3 etc.
c) f(x) = 2x3; f(0) = 0
f(x) = 3x3 ; f(0) = 0
d) Two function that satifies the equation xf' (x) = 5f(x) are f(x) = x5 and f(x) = 2x5.
e) Two functions that satisfies the equaton xf' (x) = kf(x) are f(x) = xk and f(x) = 2xk
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