Let f be the function defined by f(x) =x^4 -3x^2 +2
a)find the zeros of f
b)write an equation of the line tangent to the graph of f at the point where x=1
c) find the x coordinate of each point at which the line tangent to the graph of f is parallel to the line y=-2x+4

A convenient way to find the zeros of   is by factoring.

a) The equation,

can be rewritten as,

We can think of this equation as a quadratic equation in  , with .

Observe that .

We find two factors of 2 that adds up to . These are, .

Now let us split the middle term. to obtain;

We can factor to get,

We factor further to obtain;

Hence the zeroes of are;

b) To find the line tangent, we must first, find the slope using differentiation. That is,

At ,

Also, we need to determine the value at . That is;

Now we can use the slope and the point to write ythe equation of the line tangent.

c)

If the line tangent is parallel to the line , then

Since parallel lines have the same slope.

Hence the x-coordinates are,

step-by-step explanation:

30% of $420 is 0.30($420) = $126 and $420 + $126 = $546.

or, 130% of $420 is 1.3($420) = $546.

notice that this is not the $600 original list price with which we started.

why doesn't this work?

the method does not work because 30% of $420 is not the same amount as the 30% of $600. it may seem obvious when stated this way, but it is a very common error made on problems of this type. we cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

to work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

example: since (100% – 30%)(original price) = (sale price)

70% of the   (original price) is $420.

0.7x   = 420

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x = 600

the original cost was $600.

we may also solve this problem using the two-step method. working the problem in two steps requires us to use more algebra skills. we have to know that x = 1x (identity property of multiplicationtext annotation indicator) and that 1x – 0.3x = 0.7x (subtracting like terms). then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = $420

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x = 600

the original cost was $600.  

note that with both of the above methods we do obtain the correct original amount.

example: after a 5% pay raise, hermione is earning $22,680 per year. what was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

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x = 21,600

hermione earned $21,600 per year before the pay raise.

this is an example so maybe you will get how to at least do it!

the answer is b