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Minimum value

Problem. Find the minimum value of over Solution: Let Here the function is continuous and differentiable on Therefore For critical points , which implies , The first equation gives The second equation gives or squaring we get , for some Now we get the values of f at these critical…

Function

Problem: Show that the function given by is a one-to-one function Solution: We know that a function is a one-to-one function iff for all Case 1: Let be even integers Case 2: Let be odd integers Case 3: Let be odd and be even then and i.e. is a false…

Range of a function

Question: Find the range of the function given by . Solution: The domain of the given function is closed interval . Since the function is continuous on a closed interval, therefore it attains its bound (the greatest and least value), somewhere in the closed interval. By Fermat’s theorem if the…

Trigonometric Inequalities

Problem: Show that cos(sinx)>sin(cosx) for each real x. Sol. Consider the equation Taking + we get, which is not possible Taking – we get, which is again not possible Hence the equation has no solution. Now consider the function . The function is continuous on , therefore exactly one of…