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Question:
Consider the following statements in respect of the function $f(x)=x^{3}-1, x \in[-1,1]$
I. $f(x)$ is increasing in $[-1,1]$
II. $f(x)$ has no root in $(-1,1)$.
Which of the statements given above is/are correct?
Consider the following statements in respect of the function $f(x)=x^{3}-1, x \in[-1,1]$
I. $f(x)$ is increasing in $[-1,1]$
II. $f(x)$ has no root in $(-1,1)$.
Which of the statements given above is/are correct?
Updated On: Mar 18, 2024
- Only I
- Only II
- Both I and II
- Neither I nor II
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The Correct Option is A
Solution and Explanation
Since $f (x)$ is an increasing function in $[- 1, 1]$ and it has a root in $(- 1, 1)$.
$\therefore$ Only statement I is correct.
$\therefore$ Only statement I is correct.
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Concepts Used:
Increasing and Decreasing Functions
Increasing Function:
On an interval I, a function f(x) is said to be increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≤ f(y)
Decreasing Function:
On an interval I, a function f(x) is said to be decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≥ f(y)
Strictly Increasing Function:
On an interval I, a function f(x) is said to be strictly increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) < f(y)
Strictly Decreasing Function:
On an interval I, a function f(x) is said to be strictly decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) > f(y)
Graphical Representation of Increasing and Decreasing Functions
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