Question

Pls solve.Won't the maximum value occur at larger values of x such as 5,6...

Answer 1

maximum at (0 , 1 )
minimum at (2 , -3 )

Explanation:To find the x-coordinates of maxima / minima differentiate thefunction and equate to zero ( the gradients of tangents atstationary points = 0 ).let y = x3−3x2+1now dydx=3x2−6x=0hence 3x (x - 2 ) = 0 and x = 0 or x = 2To find y-coordinates substitute these values into y .

x = 0 : y = 1 hence (0 , 1 ) is a stationary point.( S.P )

x = 2 : y = (2)3−3(2)2+1=−3hence ( 2 , - 3 ) is an S.PTo find the 'Nature' of these points use the second derivative.If d2y/dx2 < 0 then maximum S.PIf d2y/dx2> 0 then minimum S.Pnow d2y/dx2=6x−6x=0:d2y/dx2=−6<0⇒(0,1) is a maxx=2:d2y/dx2=12−6=6>0⇒(2,−3)is a min

Answer 2

Maximum value at x=-2
y= 10
Minimum value at x=2
y=2