Figure 1 shows part of a curve C with equation $y = 2x^2 + \frac{8}{x} - 5$, for $x > 0$ - Edexcel - A-Level Maths Pure - Question 2 - 2016 - Paper 4

Find the derivative: Start with the function $y = 2x^2 + \frac{8}{x} - 5$. To find where the function is increasing, we need to determine its derivative:

$\frac{dy}{dx} = \frac{d}{dx}\left(2x^2 + \frac{8}{x} - 5\right) = 4x - \frac{8}{x^2}$

Set the derivative greater than zero: To determine where the function is increasing, we set the derivative greater than zero:

$4x - \frac{8}{x^2} > 0$

Rearranging gives:

$4x^3 > 8$

$x^3 > 2$

Solve for x: Taking the cube root of both sides results in:

$x > \sqrt[3]{2}$

Evaluation: Since $\sqrt[3]{2} \approx 1.26$, it follows that for all $x > 2$, the derivative $\frac{dy}{dx}$ will be positive, indicating that the function is indeed increasing in this interval.