Question

Suppose f(x) is continuous and positive for all x. moreover, suppose f'(x) is positive for all x.

select the appropriate comparison
∫₀⁵f(x) dx ___ 5∫₀¹ f(x)dx

Answer 1

The integral ∫₀⁵ f(x) dx is greater than 5∫₀¹ f(x) dx due to the properties of the function and its derivative.

Step-by-step explanation:

The question asks us to compare the integrals of two functions. Let's compare the two integrals step by step:

1. First, let's calculate the integral of f(x) from 0 to 5: ∫₀⁵ f(x) dx
2. Next, let's calculate the integral of f(x) from 0 to 1 and multiply it by 5: 5∫₀¹ f(x) dx
3. Since f(x) is positive and continuous, and f'(x) is positive for all x, we can conclude that f(x) is an increasing function. This means that f(x) will have a greater value when integrated over a larger interval. So, the value of ∫₀⁵ f(x) dx will be greater than 5∫₀¹ f(x) dx.

Therefore, the appropriate comparison is: ∫₀⁵ f(x) dx > 5∫₀¹ f(x) dx.