Question

find the stationary point of the curve y=-xlogx and also determine the nature of the stationary point. please show all working out

Answer 1

dy/dx = -x * 1/x + -1 logx (assuming its log to base e)
so
-1 - 1 logx = 0 at stationary points
-1 = log x
x = e^-1
stationary point at (e^-1, -0.3679)

second derivative = -1 * 1/x which is negative
stationary point is a maximum

Answer 2

sorry the point is (0.3679,0.3679)

Answer 3

or in terms of e it is (e^-1,e^-1)

Answer 4

ok with that virtus?

Answer 5

i used the product rule to differentiate

Answer 6

@ jimmyrep thanks a bunch
but could you just run through how to determine the nature of stationary points like the steps cause i am quite confused on that part
thanks

Answer 7

and can you also brief me on how to find points of inflexions
thanks

Answer 8

ok - if the second derivative is negative its a maximum positive its minimum
here the second derivative = d(-1 - 1 logx) dx = -1/x
plug in x = 0.3679 : it is -1/0.3679 which is negative so maximum

the rate of change of dy/dx going over the 'hump' in a maximum is towards the negative gradient
and with minimum its the opposite

Answer 9

ok if its a point of inflection that is horizontal then second derivative will equal zero
other points of inflection are more complicated - sorry i have no time to go into those - i have to go out.

Answer 10

oh thank you jimmyrep for the much informative explanation! =D