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Thanks ↑ laszky: But answer Given as .
i have one doubt How can we assume function in first question. (We can assume other function also like type
Similarly How can we assume function for second question. We can also assume other function.
please clearfy me , thanks in advanced.
↑ stuart clark:
The crucial words are "at least". At first you have to notice that , then you observe that for arbitrary and you choose in such a way that has maximal possible zeros. If you choose them wrong, you prove only partial result (not the optimal one) - like me in my previous answer. (I forgot that you can substract any linear function and substracted only constant). My approach is: draw a picture of the presumable graph of the function and then try to draw a line which intersects the function in maximal possible points.
The following picture shows my first attempt with 6 zeros of the function g and my second attempt with 7 zeros of the function g (due to the zero derivative of there are two intersections near the point ). Hence, using this second attempt, we can prove that there exists at least 5 roots of . But 6 is still problem for me.