Test of Mathematics Solution Subjective 75

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}$.

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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}$.