How can you show that the curve y=x - ln x has only one turning point?
Answers:
You have two approaches.
1. Select a range of values for x, determine y and then plot a graph.
2. Differentiate y = x - In x
dy/dx = 1 - 1/x
This gives the gradient at any point on the curve.
When there is a turning point the gradient of a curve = 0
So we put dy/dx = 0 or 1 - 1/x = 0
Hence 1 = 1/x, and x = 1
There is only one value hence there is only one turning point
If you need to show if the turning point is a local maximum, a local minimum of a point of inflexion use the second and third derivatives.
dy/dx=1-1/x=0
1/x=1
x=1
therefore there is only one turning point
dy/dx = 1 - 1/x
turning point when dy/dx = 0
so, 1 - 1/x = 0 => 1/x = 1, ie x = 1, only one solution, therefore one turning point. also, you could look at the graph of the function.
Differentiate it and solve for x, you will see the derivative has only one root, hence the original equation has only one turning point.
Consider a cubic function, differentiating a cubic gives a quadratic, and a quadratic has two solutions, hence a cubic has two turning points.
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