Solve the Following Question.(2 Marks)

Question 12 Marks

Write the minimum value of $\text{f}(\text{x})=\text{x}+\frac{1}{\text{x}},\text{x}>0.$

Answer

Given, $\text{f}\text{(x)}=\text{x}+\frac{1}{\text{x}}$
$\therefore\ \text{f}'\text{(x)}=1-\frac{1}{\text{x}^2}$
For a local maxima or a local minima,
$\text{f}'\text{(x)}=0$
$\Rightarrow1-\frac{1}{\text{x}^2}=0$
$\Rightarrow\text{x}= +1,-1$
Now, $\text{f}''(\text{x})=\frac{2}{\text{x}^{3}}$
At x = 1, f''(x)
x = -1, f''(x) = -2 < 0
$\therefore $ x = -1 is point of local minima
$\therefore $ maximum value of = f(-1) = -2.

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Question 22 Marks

If f(x) attains a local minimum at x = c, then write the values of f' (c) and f'' (c).

Answer

If f(x) attains a local minimum at x = c, then the first order derivative of the function at the given point must be equal to zero, i.e.
f'(x) = 0 at x = c
⇒ f'(c) = 0
The second order derivative of the function at the given point must be greater than zero, i.e.
f''(c) > 0

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Question 32 Marks

Write sufficient condition for a point x = c to be an extreme point of the function f(x).

Answer

Let f(x) be a function.
f(x) will attain its extreme value at a point x = c if f'(c) = 0
The point x = c is known as point of local maxima if f''(x) < 0

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Question 42 Marks

Write the point where $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$ attains minimum value.

Answer

Given, $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$
$\Rightarrow\text{f}'(\text{x})=\log_{\text{e}}\text{x}+1$
For a local maxima or a minima, we must have $\text{f}(\text{x})=0$
$\Rightarrow\log_{\text{e}}\text{x}+1=0$
$\Rightarrow\log_{\text{e}}\text{x}=-1$
$\Rightarrow\text{x}=\frac{1}{\text{e}}$
$\Rightarrow\text{f}\Big(\frac{1}{\text{e}}\Big)=\frac{1}{\text{e}}\ \log_{\text{e}}\Big(\frac{1}{\text{e}}\Big)=-\frac{1}{\text{e}}$
Now, $\text{f}''(\text{x})=\frac{1}{\text{x}}$
At, $\text{x}=\frac{1}{\text{e}}$
$\text{f}''\Big(\frac{1}{\text{e}}\Big)=\frac{1}{\frac{1}{\text{e}}}=\text{e}>0$
So, $\Big(\frac{1}{\text{e}},-\frac{1}{\text{e}}\Big)$ is a point of local minimum.

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Question 52 Marks

Write necessary condition for a point x = c to be an extreme point of the function f(x).

Answer

We know that at the extreme points of a function f(x), the first order derivative of the function is equal to zero, i.e.
f'(x) = 0 at x = c
⇒ f'(c) = 0

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Maths Solve the Following Question.(2 Marks)

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