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## Calculus - Differential Calculus - Problem Set III - Outline of Contents (Also try out the MCQ Quiz at the end):

### This tutorial contains examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem ; inspection of polynomials and trigonometric expressions.

Here's a quick look at the kind of problems which have been solved in this tutorial.  We will try to sketch some curves and inspect them visually as well.
• Verify Rolle’s theorem for (i)x2 in [−1, 1].    (ii)x(x + 3)e−x/2 in [−3, 0].
• Learning to apply Leibnitz formula for differentiation.
• Seperate the intervals in which the polynomial 2x3 − 15x2 + 36x + 1 is increasing or decreasing. Also draw the graph of the function.
• Show that x2 > (1 + x)[log (1 + x)]2 for x > 0.
• Find the greatest and the least values of 3x4 − 2x3 − 6x2 + 6x + 1 in the interval [0, 2].
• Examine the polynomial 10x6 − 24x5 + 15x4 − 40x3 + 108 for maximum and minimum values.
• Find the extreme values of x4 − 8x3 + 22x2 − 24x + 1 and distinguish between them.
• Find the maximum and minimum values of the polynomial 8x5 − 15x4 + 10x2 .
• Find the maximum and the minimum values of the function sin x + cos 2x.
• Find the slope and concavity of the graph of x2 y + y 4 = 4 + 2x at the point (−1, 1).
• Consider the equation x2 + xy + y 2 = 1. Find equations for y and y in terms of x and y only.
This way, you will gain an understanding of maxima and minima; be familiar with what it means for a function to be monotonically increasing or monotonically decreasing or have zeros, in a particular range. The process of finding and identifying a maxima or minima is known as optimization.

To gain a good understanding of maxima/minima, it is important to visualize and draw curves or figures. Looking at a function, one should intuitively be able to identify its zeros or the points where dy/dx = 0, in-flexion points, points of continuity and dis-continuity; points where the curve may not be differentiable.

### MCQ Quiz- Differential Calculus and Mean Value Theorems

Companion MCQ Quiz for Differential Calculus and Mean Value Theorems: test how much you know about the topic. Your score will be e-mailed to you at the address you provide.