Question:
Let $f : \left(-\infty, \infty\right) \to \left(-\infty , \infty \right)$ be defined by $f(x) = x^3 + 1$.
The function f has a local extremum at $x = 0$
The function f is continuous and differentiable on (-??,oo) and/'(0) = 0
Let $f : \left(-\infty, \infty\right) \to \left(-\infty , \infty \right)$ be defined by $f(x) = x^3 + 1$.
The function f has a local extremum at $x = 0$
The function f is continuous and differentiable on (-??,oo) and/'(0) = 0
Updated On: Jul 28, 2023
- Statement 1 is true, Statement 2 is false.
- Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
- Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.
- Statement 1 is false, Statement 2 is true.
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The Correct Option is D
Solution and Explanation
$f : \left(-\infty, \infty\right) \to \left(-\infty , \infty \right)$ be defined by $f\left(x\right) = x^{3} + 1$.
Clearly, $f\left(x\right)$ is symmetric along $y = 1$ and it has neither maxima nor minima.
$\therefore$ Statement - 1 is false.
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
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