calc question?

 Find all the values of  x  where the graph of f(x)=2x^2 -24x^2+42x-10 has a horizontal tangent line.

The smaller one is X=

And the larger one is X=

5 Answers

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  • 4 weeks ago

    Poor presentation: why not f(x)=-22x^2+42x-10?! If so,

    then f '(x)=-44x+42=0=>x=21/22 is the only solution.

    Check out your mistake & re-post your problem.

  • 1 month ago

    Several others have attempted to "correct" the equation.

    I'll do it as presented.

     

    If the given equation is wrong, the answer will be wrong.

    "Garbage In => Garbage Out"

    ---------------------------------

    f(x) = 2x² - 24x² + 42x - 10 

    Combine like terms

    f(x) = -22x² + 42x - 10 

    -------------------------

    Method I - Algebra

    This is a parabola

    Only the tangent at the vertex is horizontal

    For quadratics of the form

    y = ax²  + bx + c

    The x coordinate of the vertex is

    x = -b/(2a)

    Therefor

    x = -42 / (2 * -22)

    x = 42/44

    x = 21/22 <––––––

    -------------------------

    Method II - Calculus

    The first derivative is the slope

    f'(x) = -44x + 42

    A horizontal line has a slope of 0

    -44x + 42 = 0

    Subtract 42 from both sides

    -44x = -42

    Divide both sides by -44

    x = 22/44

    x = 21/22 <–––––

  • rotchm
    Lv 7
    1 month ago

    First, simplify your f(x); write it in the form ax² + bx + c. That is pre highschool math, so surely you can do it.

    A horizontal tangent line means  that the slope of f(x) is zero.

    Since this is a quadratic, it's slope is zero at its vertex. Again that is high school math. So surely you can find it.

    Or via calculus: 

    The derivative of f(x) = 0.

    Whats the derivative of your f(x) ?

    If you meant  2x^3  - 24x²..., same reasoning. 

    Answer that then we will take it from there if need be. 

    Hopefully no one will spoil you the answer. That would be very irresponsible of them. Too late. So many irresponsible people here  :/

    Don't forget to vote me best answer for being the first to correctly walk you through without spoiling you the explicit answers.

  • Did you mean 2x^3 - 24x^2 + 42x - 10?  If so

    f(x) = 2x^3 - 24x^2 + 42x - 10

    f'(x) = 6x^2 - 48x + 42

    f'(x) = 0

    0 = 6x^2 - 48x + 42

    0 = x^2 - 8x + 7

    0 = (x - 7) * (x - 1)

    x = 1 , 7

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  • 1 month ago

    as written, you have a parabola that will only have one horizontal tangent line (slope = 0).  So presuming you really meant:

    f(x) = 2x³ - 24x² + 42x - 10

    To get the first derivative of this polynomial expression, for each term we can perform the following steps:

    The new coefficient is the product of the old coefficient and old exponent.

    The new exponent is one less than the old exponent.

    Any terms that are first-degree become constants (as x⁰ = 1).

    Any constant terms are dropped.

    So the first derivative of your function is:

    f(x) = 2x³ - 24x² + 42x - 10

    f'(x) = 6x² - 48x + 42

    Set f'(x) to zero and solve for the two x's:

    0 = 6x² - 48x + 42

    Simplify by dividing both sides by 6:

    0 = x² - 8x + 7

    This factors:

    0 = (x - 7)(x - 1)

    So the roots are:

    x = 1 and 7

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