At which value of 𝑥 does the
function 𝑓 of 𝑥 equals 𝑥 plus three all squared plus four have its minimum?
So if we take a look at our
function, we want the part 𝑥 plus three all squared to be the lowest possible value
it can be if we want the value of our function to be as its minimum. Well, we know that 𝑥 plus three
all squared is gonna be greater than or equal to zero. And that’s because any number
squared is gonna be zero or above. So it’s gonna be zero or a positive
number. So therefore, the lowest possible
value of 𝑥 plus three all squared has got to be zero.
So before we work out which value
of 𝑥 is gonna give us this minimum, we need to decide how do we know it’s gonna be
a minimum, not a maximum. We know it’s gonna be a minimum
because if we multiply out of our parentheses 𝑥 plus three all squared, we’re gonna
have a positive 𝑥 squared term. So therefore, we know the shape of
our graph is going to be a parabola, a U-shaped parabola. If it was a negative 𝑥 squared
term, then we’d actually have an inverted U-shaped parabola.
So therefore, to find our minimum
point, what we want to do is set 𝑥 plus three all squared equal to zero cause we
want to find out what’s going to make this equal to zero. So then, if we take the square root
of each side of the equation, we’re gonna get 𝑥 plus three is equal to zero. And then, we subtract three from
each side of the equation. We’re gonna get 𝑥 is equal to
And if we put this back in, we
would have negative three plus three. Well, that’s just zero. Zero squared is equal to zero. So therefore, we can say that the
value of 𝑥 that the function 𝑓 of 𝑥 equals 𝑥 plus three all squared plus four
has its minimum is going to be 𝑥 equals negative three.
And it’s worth noting that this
minimum point is gonna have the coordinates negative three, four and that’s because
our 𝑥-value is negative three. If we have that, then 𝑓 of 𝑥
would be equal to zero plus four. So therefore, the value of 𝑦 was
gonna be four. So we’d have the coordinate
negative three, four to the minimum point.