Recognizing features of functions (example 3) Video Lecture - Engineering Mathematics

Which function has a local
minimum with a value of 1?
So let's think about
this a little bit.
So they tell us the
x values, and we
see x increasing from negative
4 all the way to positive 4.
So our x values are increasing.
And they give us
different functions here.
And they essentially say
which of these functions
have a local minimum value where
the value of the minimum is 1?
So they hit a
minimum point at 1.
So that means that the
value of the function
must be 1 at that minimum point.
So we can see candidate
points right over here.
F of x hits 1 right over there.
G of x hits 1 right over there.
And h of x hits 1
right over there.
And let's see, that's the
only time that they hit 1.
Now let's think about which
of these are minimum values.
So on f of x, when we go from
negative 2 to negative 1,
we are increasing from 0 to
1 and then we increase again.
Well, that's not going
to be a minimum value.
We are increasing.
So let me draw a little
axis right over here.
So let me actually draw
the whole thing just
to make it clear
what we're doing.
So we have negative 4, 3, 2, 1.
Positive 1, 2, 3, 4.
And of course this is 0,
negative 4, negative 3,
negative 2, negative 1.
1, 2, 3, 4, and this is going
to be 0 right over here.
And we're going to
care about what's
happening when these
functions hit 1.
So f of x, when x is
equal to negative 2, is 0.
When x is equal to
negative 1 is 1,
and then when x is
equal to 0 is 5.
So 2, 3, 4, 5 is that
right over there.
And if we assume that these are
sampling from the function--
we actually necessarily can't
make that assumption here--
but if we assume that
these are sampling, then
if this was a
continuous function,
then we would assume
that the function does
something like this.
And so it's clearly not a
minimum point right here.
We're increasing into 1.
So this is not a minimum point.
Now let's look at g of x.
Let's see, when g of
x is 1, we're at 2.
When g of x is at 2, we're at 1.
And when g of x is 3,
we're at negative 1.
So we are at negative 1.
So this is the opposite.
We do decrease into that value
when the function hits 1,
but then we keep decreasing.
In order for this to
be a minimum point,
we would have to decrease into
it and then increase out of it.
This right over here is
also not a minimum point.
Now let's look at h of x.
When x is negative
4, we're at 5.
When x is at negative
3, we are at 1.
And when x is at negative
2, we jump back up to 3.
So this looks like
a minimum point.
We decreased going
into it and then
we increased going out of it.
So this is our
local minimum point.
And if this was a
continuous function
it maybe might look
something like that.
So we will go with h of x.