Question

Identify the expression that has the largest value when a<-1 1-2a a a^2 a^3

Answer 1

Answer:

1-2a is the largest for   -1 > a > -2.41

a^2 is the largest for  -2.41 > a.

Step-by-step explanation:

We want to find the expression with the largest value.

Remember that the sign is important here, because:

1 is larger than -10

So we will look at the expressions that, when evaluated in a negative number, give a positive number.

The only two options in the case:

a < -1

are:

1-2a

and

a^2

So we have a linear equation and a quadratic equation.

We know that quadratic equations grow a lot faster than linear equations for all values larger than 1 (or minus one, because the sign does not matter).

if -1 < a < 1, a quadratic function grows really slow, but this is not the case.

So from now, we can guess that a^2 is larger.

But let's solve it graphically.

Let's define two functions:

f(a) = 1 - 2*a

g(a) = a^2

Now we can graph these two, and look at the region a < -1, and see which one is larger.

in the image, you can see that there is a small region from:

-1 > a > -2.41

Where the linear equation is larger, and for:

a < -2.41

The quadratic equation is larger.