## Q. 12.4

Locate all maximum points, minimum points and points of inflexion of  $y=x^4$.

## Verified Solution

$y^{\prime}=4 x^3 \quad y^{\prime \prime}=12 x^2$

y′ = 0 at x = 0. Also y′′ = 0 at x = 0 and so the second-derivative test is of no help in determining the position of maximum and minimum points. We return to examine y′ on both sides of x = 0. To the left of x = 0, y′ < 0; to the right y′ > 0 and so x = 0 is a minimum point. Figure 12.12 illustrates this. Note that at the point x = 0, the second derivative y′′ is zero. However, y′′ is positive both to the left and to the right of x = 0; thus x = 0 is not a point of inflexion.