**Calculus: Graphical, Numerical, Algebraic, 3rd Edition Answers Ch 4 Applications of Derivatives Ex 4.1**

Calculus: Graphical, Numerical, Algebraic Answers

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There is no point, where graph is parallel to x-axis. Hence no critical point is possible in this case.

(b) The number of critical points simply gives us the number of values of x where the graph of the function has a tangent parallel to the x-axis. These points are only probable candidates to become local extreme values of f. However, they do not always represent the local extreme values.

The first derivative is put to zero in order to identify the critical points. The second derivative is then found to identify if that critical point is the local minima (when second derivative is positive) or the local maxima (when second derivative is negative). In the case when the second derivative is zero, then it is neither maxima nor minima. It then becomes the point of inflexion.

Consider the examples taken in part (a).

In the first example, first derivative gave 2 critical points; one was identified to be a maxima and the other was identified to be a minima.

In the first example, first derivative gave 1 critical point; this one was identified to be neither maxima nor minima. This is so because the second derivative of the function at x = 0 is equal to zero. This point is called the point of inflexion.

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The given function has no local extreme value at x = 1. It is a continuous function on [0, 1].