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).

(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.

... (1) When b^2-3ac>0 there are two stationary points (2) When b^2-3ac=0 there is exactly one stationary point (3) When b^2-3ac<0 there are no stationary points. ...

... b) i) Applying First Derivative Test, classify left hand stationary point in part a. ii) Applying First Derivative Test, classify right hand stationary point...

... the Stationery Stop is explored. http://www.microsoft.com/businesssolutions/ retailmanagementsystem/product/product_c omparison.mspx. Compare Microsoft point- ...

... And finally how would I locate and classify all the stationary points of f? ... So we classify both stationary points (-4, 6) and (4, 6) are saddle points. ...