If 1 < x < 9/8, which is bigger, ∛3x or √2x.
at the lower end , that is, at x = 1
3^(1/3) = 1.44..
√2 = 1.414.. so ∛3x > √2x
at x = 9/8
(27/8)^(1/3) = 3/2 = 1.5
√(18/8) = √(36/16) = 6/4 = 1.5
so for all values for 1 < x < 9/8
∛3x > √2x
They become equal at x = 9/8
as a matter of fact, ∛3x > √2x for 0 < x < 9/8
http://www.wolframalpha.com/input/?i=plot+y+%3D+%E2%88%9B(3x)+,+y+%3D+%E2%88%9A(2x)
To determine which is bigger between ∛3x and √2x when 1 < x < 9/8, we can compare the two expressions by substituting different values of x within the given range and observing their values.
Let's first consider ∛3x:
∛3x = (∛3)(∛x)
Now, plug in the lower bound of x, which is 1:
∛3(1) = ∛3
Now, let's consider √2x:
√2x = (√2)(√x)
Again, substitute the lower bound of x, which is 1:
√2(1) = √2
So far, we have ∛3 and √2 as the values of ∛3x and √2x when x = 1.
Next, let's consider the upper bound of x, which is 9/8:
For ∛3x:
∛3(9/8) = ∛(27/8) = ∛(3/2) = ∛(√2/2)
And for √2x:
√2(9/8) = √(18/8) = √(9/4) = √(9/2) = √(4.5)
Now we have ∛(√2/2) and √(4.5) as the values for ∛3x and √2x when x = 9/8.
Comparing the values we obtained:
∛3x when x = 1: ∛3
√2x when x = 1: √2
And:
∛3x when x = 9/8: ∛(√2/2)
√2x when x = 9/8: √(4.5)
Since it's challenging to determine which expression is bigger just by comparing the values directly, we can rely on a calculator or a mathematics software to find the exact numerical values. We can input the expressions ∛(√2/2) and √(4.5) into a calculator or mathematical software.
After doing this, we find that approximately:
∛(√2/2) ≈ 0.8409
√(4.5) ≈ 2.1213
Based on these results, we can conclude that √2x is bigger than ∛3x when 1 < x < 9/8.