# Question 12.5: Find any maximum points, minimum points and points of inflex......

Find any maximum points, minimum points and points of inflexion of y = x³ + 2x².

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.

Given y = x³ + 2x² then y′ = 3x² + 4x and y′′ = 6x + 4. Let us first find any maximum and minimum points. The first derivative y′ is zero when 3x² + 4x = x(3x + 4) = 0, that is when x = 0 or  $x=-\frac{4}{3}$.  Using the second-derivative test we nd y′′(0) = 4 which corresponds to a minimum point. Similarly,  $y^{\prime \prime}\left(-\frac{4}{3}\right)=-4$  which corresponds to a maximum point.

We seek points of inflexion by looking for points where y′′ = 0 and then examining the concavity on either side. y′′ = 0 when  $x=-\frac{2}{3}$.

Since y′′ is negative when  $x<-\frac{2}{3},$  then y′ is decreasing there, that is the function is concave down. Also, y′′ is positive when  $x>-\frac{2}{3}$  and so y′ is then increasing, that is the function is concave up. Hence there is a point of inflexion when  $x=-\frac{2}{3}$.  The graph of y = x³ + 2x² is shown in Figure 12.13.

Question: 12.7

## A root of 3 sin x = x is near to x = 2.5. Use two iterations of the Newton–Raphson technique to find a more accurate approximation. ...

The equation must first be written in the form f (...
Question: 12.8

Question: 12.1

## Determine the position and nature of all maximum and minimum points of the following functions: (a) y = x² (b) y = −t² + t +1 (c) y = x³ / 3 + x² / 2 − 2x + 1 (d) y  = |t| ...

(a) If y = x², then by differentiation  \fr...
Question: 12.6

## Given that x1 = 7.5 is an approximate root of e^x − 6x³ = 0, use the Newton–Raphson technique to find an improved value. ...

\begin{aligned}x_1 & =7.5 & & \...
Question: 12.10

## If a = 3ti − t²j and b = 2t²i + 3j, verify  (a) d / dt (a • b) = a • db / dt + da / dt • b  (b) d / dt (a × b) = a × db / dt + da / dt × b ...

(a) \begin{gathered}\mathbf{a} \cdot \mathb...
Question: 12.9

## If a = 3t²i + cos 2tj, find (a) da / dt  (b) |da / dt| (c) d²a / dt² ...

(a) If  \mathbf{a}=3 t^2 \mathbf{i}+\cos 2 ...
Question: 12.4

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

y^{\prime}=4 x^3 \quad y^{\prime \prime}=12...
Question: 12.3