when does this function stop growing?

Question

Hello to all the folks,

Let $z=0.22$

i want to know if this function never stops growing?

$$y= 150 * x^z $$

the second derivative is: $$\frac{d^2y}{dx^2}= \frac{150\cdot z\cdot(z-1)}{x^{z+1}} $$

this function can never be zero

Thank you very much

Answer 1

Consider the line through $(x,y) = (x,150 x^z)$, tangent to the function $y = 150 x^z$. For any $x$, there is a unique $y$ for the point on the function, so there is a unique tangent line through that point and therefore a unique slope of the tangent line for each $x$. The slope of that tangent line is given by the first derivative: $$ y'(x) = 150 z x^{z-1} \text{.} $$

For $x > 0$, we see that $150 > 0$, $z = 0.22 > 0$, and $x^{z-1} > 0$. Therefore, the derivative is the product of three positive numbers, so is positive. Since the derivative is positive, the function is increasing for every $x > 0$. So as we follow this function to the right, it forever increases.