Answer
438k+ views
Hint: We know that $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ both are hyperbolic functions with $ \cosh \left( x \right)=\dfrac{{{e}^{x}}+{{e}^{-x}}}{2} $ and $ {{\cosh }^{-1}}\left( x \right)=\ln \left( x+\sqrt{{{x}^{2}}-1} \right) $ . Now, find the minimum value of $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ from the equation that we have just described by putting $ x=0 $ in these equations and then compare the minimum values from the graph shown in the above problem. If the minimum values of $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ from the equations are matched with the graph given above, then the curves are correctly labeled.
Complete step-by-step answer:
The below graph shown in the above question has two curves marked with red and green colour.
It is given that green color curve corresponds to $ \cosh \left( x \right) $ and red color corresponds to $ {{\cosh }^{-1}}\left( x \right) $ and we have to show whether the labeling of the curves are correct or not.
We know that $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ are hyperbolic functions. In the below, we are showing the hyperbolic functions in x corresponding to $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ .
The function of $ \cosh \left( x \right) $ is equal to:
$ \cosh \left( x \right)=\dfrac{{{e}^{x}}+{{e}^{-x}}}{2} $ …………. Eq. (1)
The function of $ {{\cosh }^{-1}}\left( x \right) $ is equal to:
$ {{\cosh }^{-1}}\left( x \right)=\ln \left( x+\sqrt{{{x}^{2}}-1} \right) $ ……….. Eq. (2)
Now, to check whether the curves labeled in the above question are correct or not by finding the minimum values of $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ .
Substituting $ x=0 $ in eq. (1) we get,
$ \begin{align}
& \cosh \left( 0 \right)=\dfrac{{{e}^{\left( 0 \right)}}+{{e}^{\left( 0 \right)}}}{2} \\
& \Rightarrow \cosh \left( 0 \right)=\dfrac{1+1}{2} \\
& \Rightarrow \cosh \left( 0 \right)=\dfrac{2}{2}=1 \\
\end{align} $
Now, at $ x=0 $ we have got the value of $ \cosh \left( x \right) $ as 1 which is the same as given in the above problem.
Hence, the green curve corresponding to $ \cosh \left( x \right) $ is correctly labeled.
If we assume that what is given in the question is right i.e. red curve corresponds to $ {{\cosh }^{-1}}\left( x \right) $ then the minimum value of this function occurs at $ x=1 $ and is 0 so let us substitute $ x=1 $ in eq. (2) to see whether the value of $ {{\cosh }^{-1}}\left( x \right) $ is coming 0 or not.
$ \begin{align}
& {{\cosh }^{-1}}\left( x \right)=\ln \left( x+\sqrt{{{x}^{2}}-1} \right) \\
& \Rightarrow {{\cosh }^{-1}}\left( 1 \right)=\ln \left( 1+\sqrt{{{1}^{2}}-1} \right) \\
\end{align} $
$ \begin{align}
& \Rightarrow {{\cosh }^{-1}}\left( 1 \right)=\ln \left( 1+\sqrt{0} \right) \\
& \Rightarrow {{\cosh }^{-1}}\left( 1 \right)=\ln \left( 1 \right) \\
\end{align} $
And we know that, the value of $ \ln \left( 1 \right)=0 $ so the value of the above equation becomes:
$ {{\cosh }^{-1}}\left( 1 \right)=0 $
Hence, we have got the same minimum value of $ {{\cosh }^{-1}}\left( x \right) $ which is given in the above problem. Hence, the red curve which is labeled as $ {{\cosh }^{-1}}\left( x \right) $ is correct.
From the above, we can say that the above statement is true.
Note: Don’t confuse the functions $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ given in the above problem with $ \cos x\And {{\cos }^{-1}}\left( x \right) $ . These two functions are completely different from each other. The functions $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ are hyperbolic functions whereas $ \cos x\And {{\cos }^{-1}}\left( x \right) $ are trigonometric functions.
Usually, students ignore the “h” written with cosine and think it might be a typo but it is not so make sure you don’t repeat such mistakes.
Complete step-by-step answer:
The below graph shown in the above question has two curves marked with red and green colour.
![seo images](https://www.vedantu.com/question-sets/319bd817-6893-4c7a-97d2-41d97b7df5f07188288188934996139.png)
It is given that green color curve corresponds to $ \cosh \left( x \right) $ and red color corresponds to $ {{\cosh }^{-1}}\left( x \right) $ and we have to show whether the labeling of the curves are correct or not.
We know that $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ are hyperbolic functions. In the below, we are showing the hyperbolic functions in x corresponding to $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ .
The function of $ \cosh \left( x \right) $ is equal to:
$ \cosh \left( x \right)=\dfrac{{{e}^{x}}+{{e}^{-x}}}{2} $ …………. Eq. (1)
The function of $ {{\cosh }^{-1}}\left( x \right) $ is equal to:
$ {{\cosh }^{-1}}\left( x \right)=\ln \left( x+\sqrt{{{x}^{2}}-1} \right) $ ……….. Eq. (2)
Now, to check whether the curves labeled in the above question are correct or not by finding the minimum values of $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ .
Substituting $ x=0 $ in eq. (1) we get,
$ \begin{align}
& \cosh \left( 0 \right)=\dfrac{{{e}^{\left( 0 \right)}}+{{e}^{\left( 0 \right)}}}{2} \\
& \Rightarrow \cosh \left( 0 \right)=\dfrac{1+1}{2} \\
& \Rightarrow \cosh \left( 0 \right)=\dfrac{2}{2}=1 \\
\end{align} $
Now, at $ x=0 $ we have got the value of $ \cosh \left( x \right) $ as 1 which is the same as given in the above problem.
Hence, the green curve corresponding to $ \cosh \left( x \right) $ is correctly labeled.
If we assume that what is given in the question is right i.e. red curve corresponds to $ {{\cosh }^{-1}}\left( x \right) $ then the minimum value of this function occurs at $ x=1 $ and is 0 so let us substitute $ x=1 $ in eq. (2) to see whether the value of $ {{\cosh }^{-1}}\left( x \right) $ is coming 0 or not.
$ \begin{align}
& {{\cosh }^{-1}}\left( x \right)=\ln \left( x+\sqrt{{{x}^{2}}-1} \right) \\
& \Rightarrow {{\cosh }^{-1}}\left( 1 \right)=\ln \left( 1+\sqrt{{{1}^{2}}-1} \right) \\
\end{align} $
$ \begin{align}
& \Rightarrow {{\cosh }^{-1}}\left( 1 \right)=\ln \left( 1+\sqrt{0} \right) \\
& \Rightarrow {{\cosh }^{-1}}\left( 1 \right)=\ln \left( 1 \right) \\
\end{align} $
And we know that, the value of $ \ln \left( 1 \right)=0 $ so the value of the above equation becomes:
$ {{\cosh }^{-1}}\left( 1 \right)=0 $
Hence, we have got the same minimum value of $ {{\cosh }^{-1}}\left( x \right) $ which is given in the above problem. Hence, the red curve which is labeled as $ {{\cosh }^{-1}}\left( x \right) $ is correct.
From the above, we can say that the above statement is true.
Note: Don’t confuse the functions $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ given in the above problem with $ \cos x\And {{\cos }^{-1}}\left( x \right) $ . These two functions are completely different from each other. The functions $ \cosh \left( x \right)\And {{\cosh }^{-1}}\left( x \right) $ are hyperbolic functions whereas $ \cos x\And {{\cos }^{-1}}\left( x \right) $ are trigonometric functions.
Usually, students ignore the “h” written with cosine and think it might be a typo but it is not so make sure you don’t repeat such mistakes.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)