WEBVTT
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Hello.
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Welcome to the exercise of the
graph of a function practice
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step.
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We have to prove that the
functions f x to x cubed,
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g x to 2 times x cubed,
h x to 1 over 2 x cubed,
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are strictly increasing and
odd, while the functions f
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prime x to minus x cubed,
g prime x to minus 2
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times x cubed, and h
prime x to minus 1 over 2
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x cubed are strictly
decreasing and odd.
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You have seen in
the Francis video
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that the function f,
which sends x to x cubed,
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is strictly increasing.
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That is, if x is less
than y, we have that x
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cubed is less than y cubed.
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Now, let us consider two cases.
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If a is any real number greater
than 0 and x is less than y,
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then we know that x cubed
is less than y cubed.
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And multiplying on both sides
by the positive real number a,
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we get that a x cubed
is less than a y cubed.
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Therefore, considering a
equal to 2 or to 1 over 2,
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we get that both g and h
are strictly increasing.
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On the other hand, if a is
a real number less than 0,
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then what we get if we will
repeat the similar reasoning,
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we get x less than y implies
x cubed less than y cubed.
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But now multiplying by a
negative number on both sides,
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this equality reverses, and
we get a x cubed greater
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than a y cubed.
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Therefore, considering a
equal to minus 1 or minus 2
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or minus 1 over 2, we get that
the functions f prime, g prime,
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and h prime are
strictly decreasing.
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And now let us examine if
these functions are odd or not.
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What means to be
an odd function.
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We have for f, g, and
h, we have to check
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which is the image of minus x.
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Precisely, f of minus
x is equal to minus
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x to the cube, which is equal
to minus x to the cube, which
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is equal to minus f of x.
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Therefore, the
function f is odd.
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Similarly, g of minus
x is equal to 2 times
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minus x to the cube, which is
equal to minus 2 x to the cube,
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which is equal to minus g of x.
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And then also g is odd.
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Now, h of minus x
is equal to 1 over 2
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minus x to the cube, which
is equal to minus 1 over 2
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x cubed, which is minus h of x.
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And also h is an odd function.
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What about now f prime,
g prime, and h prime?
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Let us compare, again,
f prime of minus
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x with minus f prime of x
and the same for the others.
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Then we have f
prime of minus x is
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equal to minus minus
x to the cube, which
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is equal to x to the cube.
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And what is minus f prime of x?
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This is equal to minus--
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f prime of x is minus x cubed,
which is equal to x cubed.
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Therefore, these two
are equal, and then
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f prime is an odd function.
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Now, g prime of minus
x is equal to minus 2
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minus x to the cube, which
is equal 2 times x cubed.
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And minus g prime of x is equal
to minus minus 2x cubed, which
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is equal to 2x cubed.
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Again, g prime of 1 minus x is
equal to minus g prime of x.
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And then g prime
is an odd function.
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Finally, h prime of minus x
is equal to minus 1 over 2
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minus x to the cube, which
is equal to 1 over 2 x
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to the cube.
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And minus h prime of x
is equal to minus minus 1
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over 2 x cubed, which is
equal to 1 over 2 x cubed.
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Again, h prime of minus x and
minus h prime of x coincide
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and therefore, also h
prime is an odd function.
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