I. The function attains maximum value.
II. The function attains minimum value.
Which of the statements given above is/are correct ?
Function Analysis: $f(x) = x$
The function given is $f(x) = x$. This is a simple linear function that increases as $x$ increases.
Interval Consideration: Open Interval $(-1, 1)$
The interval specified is $(-1, 1)$. This is an open interval, meaning it includes all numbers between -1 and 1, but not -1 and 1 themselves. Mathematically, $x \in (-1, 1)$ means $-1 < x < 1$.
Statement I: Maximum Value Attainment
In the interval $(-1, 1)$, the values of $f(x) = x$ range strictly between -1 and 1.
- As $x$ gets closer and closer to 1 (e.g., 0.9, 0.99, 0.999), $f(x)$ gets closer to 1.
- However, since $x$ must be less than 1, $f(x)$ can never be exactly equal to 1.
- Therefore, the function does not attain a maximum value within the open interval $(-1, 1)$. The supremum is 1, but it is not reached.
Statement II: Minimum Value Attainment
Similarly, in the interval $(-1, 1)$:
- As $x$ gets closer and closer to -1 (e.g., -0.9, -0.99, -0.999), $f(x)$ gets closer to -1.
- However, since $x$ must be greater than -1, $f(x)$ can never be exactly equal to -1.
- Therefore, the function does not attain a minimum value within the open interval $(-1, 1)$. The infimum is -1, but it is not reached.
Conclusion
Since neither statement I nor statement II is correct, the correct option is that the function attains neither its maximum nor its minimum value in the given open interval.
Correct Answer: Neither I nor II
Important Questions from Maxima and Minima
Four small squares of side x are cut out of a square of side 12 cm to make a tray by folding the edges. What is the value of x so that the tray has the maximum volume?
If 3 ≤ x ≤ 10 and 5 ≤ y ≤ 15 , then maximum value of \(\left(\frac{x}{y}\right)\) is-
If N is a four digit number formed by digits x 1, x 2, x 3and x 4, then maximum value of \(\frac{N}{x_{1}+x_{2}+x_{3}+x_{4}}\) is-
A wire of length 20 cm is to be bent into a rectangle. Which of the following statements is/are correct?
I. The rectangle of the largest area is the square.
II. It is possible to form a rectangle of an area of $27 \, \text{cm}^2$.Select the answer using the code given below.
Consider the following statements :
Statement-I :
The function $f(x) = \frac{x^3 + 128}{x}$ has a minimum value 48 at $x = 4$.Statement-II :
As $x$ increases through 4, $f'(x)$ changes sign from positive to negative.Which one of the following is correct in respect of the above statements?
