(i) Let http://www.maths.manchester.ac.uk/~j...c/100/0046.pnghttp://www.maths.manchester.ac.uk/~j...r/100/0028.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0078.pnghttp://www.maths.manchester.ac.uk/~j...r/100/003B.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0061.pnghttp://www.maths.manchester.ac.uk/~j...r/100/0029.pnghttp://www.mathhelpforum.com/math-help/blank be a smooth 1-parameter family of functions on http://www.maths.manchester.ac.uk/~j...r/100/0052.pnghttp://www.mathhelpforum.com/math-help/blank , with http://www.maths.manchester.ac.uk/~j...c/100/0066.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...r/071/0030.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...r/100/0028.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0078.pnghttp://www.maths.manchester.ac.uk/~j...r/100/0029.pnghttp://www.maths.manchester.ac.uk/~j...r/100/003D.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0078.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...r/071/0032.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.mathhelpforum.com/math-help/blank . Use the implicit function theorem to show there are neighbourhoods http://www.maths.manchester.ac.uk/~j...c/100/0055.pnghttp://www.mathhelpforum.com/math-help/blank and http://www.maths.manchester.ac.uk/~j...c/100/0056.pnghttp://www.mathhelpforum.com/math-help/blank of 0 such that for each http://www.maths.manchester.ac.uk/~j...c/100/0061.pnghttp://www.maths.manchester.ac.uk/~j...r/100/2208.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0055.pnghttp://www.mathhelpforum.com/math-help/blank the function http://www.maths.manchester.ac.uk/~j...c/100/0066.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...c/071/0061.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.mathhelpforum.com/math-help/blank has a unique critical point in http://www.maths.manchester.ac.uk/~j...c/100/0056.pnghttp://www.mathhelpforum.com/math-help/blank .
(ii) Consider the family http://www.maths.manchester.ac.uk/~j...c/100/0046.pnghttp://www.maths.manchester.ac.uk/~j...r/100/0028.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0078.pnghttp://www.maths.manchester.ac.uk/~j...r/100/003B.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0061.pnghttp://www.maths.manchester.ac.uk/~j...r/100/0029.pnghttp://www.maths.manchester.ac.uk/~j...r/100/003D.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0078.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...r/071/0032.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...r/100/002B.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0061.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0078.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...r/071/0033.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.mathhelpforum.com/math-help/blank . Show that for http://www.maths.manchester.ac.uk/~j...c/100/0061.pnghttp://www.maths.manchester.ac.uk/~j...r/100/2260.pnghttp://www.maths.manchester.ac.uk/~j...r/100/0030.pnghttp://www.maths.manchester.ac.uk/~j...r/100/002C.pnghttp://www.maths.manchester.ac.uk/~j...c/100/0066.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.maths.manchester.ac.uk/~j...c/071/0061.pnghttp://www.mathhelpforum.com/math-help/blankhttp://www.mathhelpforum.com/math-help/blank has more than one critical point. Explain briefly why this does not contradict the statement in part (i) of this question.
