Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 75 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


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Problem

Show that there is at least one value of {x} for which {\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} = 1}


Solution


{\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} = 1}
{\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} - 1} is a continuous function. Now if we can chose {\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} - 1} takes both negative & positive numbers then the {\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} - 1 = 0} have a solution.

For {\displaystyle{x = {\frac{1}{64}}}}, {\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} - 1 < 0} & for {x = 64}, {\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} - 1 > 0}.
So by intermediate value theorem we can say that {\displaystyle{\sqrt[3]{x}} + {\sqrt{x}} - 1 = 0} has a solution for some real {x}.