I need help with a calculus problem please?

If 10x ≤ g(x) ≤ 5×4 − 5×2 + 10 for all x, evaluate

Answer 1

This is an example of the Squeeze Theorem. Let f(x) = 10x and h(x) = 5x⁴ – 5x² + 10. Now we can write:

f(x) ≤ g(x) ≤ h(x)

Given the above relation, the Squeeze Theorem tells us that if:

k = lim f(x) = lim h(x)
            x→a             x→a

then

lim g(x) = k
x→a 

Since f(x) and h(x) are polynomials, we can evaluate these limits using direct substitution. Letting a = 1, we have:

lim f(x) = 10(1) = 10
x→1

and

lim h(x) = 5(1)⁴ – 5(1)² + 10 = 10
x→1

So now that we’ve shown

lim f(x) = 10 = lim h(x)
x→1                             x→1

we can say by the Squeeze Theorem

lim g(x) = 10
x→1

Answer 2

this is an application of what’s called the squeeze theorem.

It says that if one function (i.e. g(x) ) is between 2 other functions, and the limit of the 2 other functions is the same, then the limit of the function in the middle is also the same.

basically.

lim x–>1 of 10 x = 10 * 1 = 10.

lim x–>1 of 5x^4 – 5x^2 + 10 = 5*1 – 5*1 + 10 = 10.

therefore, lim x–>1 of g(x) is also 10.

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