# Stationary point

This question concerns the function f(x)=x^3 +3x^2 −24x+40.

(a) Find the stationary points of this function.

(b) (i) Using the strategy to apply the First Derivative Test, classify the left-hand stationary point found in part (a).

(ii) Using the Second Derivative Test, classify the right-hand stationary point found in part (a).

(c) Find the y-coordinate of each of the stationary points on the graph of the function f (x), and also evaluate f (0).

(d) Hence draw a rough sketch of the graph of the function f(x).

https://brainmass.com/math/graphs-and-functions/stationary-point-545486

#### Solution Preview

(a)

f(x) = x^3 +3x^2 −24x+40

df/dx = 3x^2 + 6x - 24

For stationary points,

df/dx = 0

=> 3x^2 + 6x - 24 = 0

=> x^2 + 2x - 8 = 0

=> x^2 + 4x - 2x - 8 = 0

=> x (x + 4) - 2 (x + 4) = 0

=> (x+4) (x - 2) = 0

=> x = -4, 2

Hence, stationary points are: -4, 2

(b)(i)

First Derivative test: Find the slopes of the function near the stationary points (both the sides).

Left side of x = -4, x = -4-h, where h is a positive small fractional (<1) number

df/dx|(x : -4 -h) = 3(-4-h)^2 + 6(-4-h) - 24

= 3(16 ...

#### Solution Summary

For given function, stationary point is obtained. Followed by using first and second derivatives, left hand and right hand are classified. Finally y co-ordinate of the stationary point is obtained and the functions is drawn.

find out the stationary point, saddle point, min and max

1) Let f(x,y)=25e^(-1/5x^2)-y^5+5y+3

a) Find all stationary points of the function f(x,y) and enter their coordinates by "" with at least 3 decimal places.

b)Let (xs,ys) be the saddle point of the function f(x,y). Calculate the following expression:

f(xs,ys)-(xs+ys) and enter the value with at least 3 dp.

c) what is(are) type of stationarity the other point(s)?

2)Let f(x,y)=6e^(x^2)-y^2+5y+6

a) stationary points

b) let (xs,ys) be the saddle point of the function f(x,y), calculate the following expression f(xs,ys)-(xs+ys)

c)what type of stationary points (stationary, saddle, min, max).