Calculus..Need Help Soon

posted by .

If f(x)=1-x^3 and f^-1 is the inverse of f, how many solutions does the equation f(x)=f^-1(x) have?

a)none b)one c)three d)five e)six

  • Calculus..Need Help Soon -

    Work out the inverse of f(x), which should be:
    f-1(x) = (1-x)(1/3)
    Equate
    f(x)=f-1(x) to get
    (1-x³)=(1-x)(1/3)
    Take cube on both sides:
    (1-x³)³ = (1-x)
    Expand and cancel "1" on each side to give
    x-3x³+3x6-x9=0 ...(1)

    This is where it gets a little fussy.
    Examine the signs of the coefficients equation and apply des Cartes rule of signs to get a minimum of three positive roots corresponding the three changes of sign.

    In addition, we know that zero is a root which is non-positive. So the total number of roots (by the des Cartes rule) is 3+2k, where k is at least 1 (because zero is a root), meaning that the number of roots is either 5, 7 or 9. Of these, only 5 is a given choice.

    I hope someone else could give a more vigorous solution.

    (The solution to equation one actually yields 5 real and 4 complex zeroes).

  • Calculus.-corr -

    Typo : "rigorous" instead of vigorous.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Algebra!!

    Six pears and three apples cost $3.90. Two pears and five apples cost $3.30. How much does one pear cost?
  2. Help!Math....

    Six pears and three apples cost $3.90. Two pears and five apples cost $3.30. How much does one pear cost?
  3. Algebra Help Inverse One-to-one

    I need to know if I have done this correctly. Problem If f is one-to-one, find an equation for its inverse. f(x)=x^3-5 y=x^3-5 x=y^3-5 5+x=y^3-5+5 5+x=y^3 My answer: y^3=5+x 3ãy=3ã5+x y=3ã5+x f^-1(x)3ã5+x
  4. Chemistry

    (1) Which of the following clusters of orbitals would form the shape trigonal bipyramidal and would also be possible within the valence shell of an atom?
  5. Calculus..Need Help Soon

    If the solutions of f(x)=0 are 1 and -2, then the solutions of f(2x)=0 are a)1 and 2 b) 1/2 and -1 c) 2 and -4 d)-2 and 4 2)-1/2 and -1
  6. Math: Calculus

    The equation: 10(x-1)(x-2)(x-3)=1, has three real solutions a<b<c where a=____, b=___ , and c=___. Enter your answers with at least six correct digits beyond the decimal point. Hint: Ask what the solutions are if the right hand …
  7. Calculus

    The equation 10(x-1)(x-2)(x-3) = 1 has three real solutions a<b<c where a = , b = , and c = . Enter your answers with at least six correct digits beyond the decimal point. Hint: Ask what the solutions are if the right hand side …
  8. Linear Algebra

    I need help please!! How many solutions does the system of equations have?
  9. Equation help 2 questions

    How many real number solutions does the equation have y=-4x^2+7x-8 none one two infinite How many real number solutions does the equation have y=3x^2+18x+27 none one two infinite Thank you!
  10. Math

    1.How many solutions does the system of equations have?

More Similar Questions