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  1. MHB Apprentice

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    #1
    Suppose that
    f(x) = (x^2 + 10)(4 - x^2).

    (A) Find all critical values of f.Critical value(s) =
    (B) Use interval notation to indicate where f(x) is increasing. Increasing: =
    (C) Use interval notation to indicate where f(x) is decreasing. Decreasing: =
    D) Find the x-coordinates of all local maxima of f. Local Maxima:=
    E) Find the x-coordinates of all local minima of f. Local Minima:=
    (F) Use interval notation to indicate where f(x) is concave up. Concave up:=
    (G) Use interval notation to indicate where f(x) is concave down. Concave down:=

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    #2
    I've taken the liberty of retitling your thread so that it describes the nature of the problem being discussed. This helps with searches and site organization in general.

    We ask that our users show what they have tried so far so we know how best to help.

    You are being asked about intervals of increasing/decreasing (slope) of a given function. What do we need to be able to analyze a function's slope?

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    #3 Thread Author
    Yes curve sketching

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    #4
    Quote Originally Posted by omwattie View Post
    Yes curve sketching
    Well, yes...answering the given questions about the function will be a good aid in sketching its graph, but let's start with part (A). We are asked to identify the function's critical values. How do we go about finding a function's critical values?

    I am asking to see how much you know about what needs to be done.

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    #5 Thread Author
    Sure I have no problem, well first of all you find the derivative of the originally equation, which I did but the answers is wrong.

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    #6
    Yes, to find the critical values, we first need to find the first derivative and then equate this derivative to zero, and solve for $x$ to get the critical values. Let's look at the function:

    $ \displaystyle f(x)=\left(x^2+10\right)\left(4-x^2\right)$

    Now, we have a couple of choices here...we can leave the function as is and use the product/power rules, or we can expand the function and then compute the derivative using the power rule on each term.

    Can you show what you did to get the derivative so I can see what you may have done incorrectly?

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    #7 Thread Author
    I used the quotient rules which gives me
    (4-x^2)(2x)-(x^2+10)(-2x)/(4-x^2)^2=28x/(4-x^2)^2
    CV=-28,-2,0,2,16

    I also did it another way by multiplied the two equations but the answers look weird

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    #8
    Quote Originally Posted by omwattie View Post
    I used the quotient rules which gives me
    (4-x^2)(2x)-(x^2+10)(-2x)/(4-x^2)^2=28x/(4-x^2)^2
    CV=-28,-2,0,2,16

    I also did it another way by multiplied the two equations but the answers look weird
    In your first post, you state:

    $ \displaystyle f(x)=\left(x^2+10\right)\left(4-x^2\right)$

    Is it actually supposed to be:

    $ \displaystyle f(x)=\frac{x^2+10}{4-x^2}$?

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    #9 Thread Author
    yeah but the equation are multiplying not dividing.

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    #10
    Quote Originally Posted by omwattie View Post
    yeah but the equation are multiplying not dividing.
    In your post (#7 in this thread), where you showed your differentiation of the function, you applied the quotient rule...leading me to suspect the function is a quotient, rather than a product as you original gave. So, assuming we actually have:

    $ \displaystyle f(x)=\left(x^2+10\right)\left(4-x^2\right)$

    Then, as I stated in post #6, we can wither apply the product/power rules, or expand the function and apply the power rule to each term. Can you choose one of these options and compute $f'(x)$?

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