A positive curve is shown on axes labelled x, y and O, symmetric about the y-axis with a peak on that axis and tails approaching the x-axis.

The diagram shows the curve with the equation $y=11+kx2\displaystyle y = \frac{1}{1+kx^2}$ , where $k\,$ is a positive constant.

Find $d2ydx2\displaystyle \frac{d^2y}{dx^2}$

Hence find the value of $k\,$ for which the curve is concave when $−36<x<36\displaystyle -\frac{\sqrt{3}}{6}<x<\frac{\sqrt{3}}{6}$ .

A positive curve is shown on axes labelled x, y and O, symmetric about the y-axis with a peak on that axis and tails approaching the x-axis.

The diagram shows the curve with the equation $y=11+kx2\displaystyle y = \frac{1}{1+kx^2}$ , where $k\,$ is a positive constant.

Find $d2ydx2\displaystyle \frac{d^2y}{dx^2}$

Hence find the value of $k\,$ for which the curve is concave when $−36<x<36\displaystyle -\frac{\sqrt{3}}{6}<x<\frac{\sqrt{3}}{6}$ .

Differentiation

Differentiation Questions