(D) For Statement-$I$: We examine the behavior of $f(x)$ near $x = 0$.For $x < 0$,$f(x) = -\frac{x}{2}$. As $x \to 0^-$,$f(x) \to 0$.For $x \geq 0$,$f

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Statement-$I$ is false,Statement-$II$ is true.

(D) For Statement-$I$: We examine the behavior of $f(x)$ near $x = 0$.For $x For $x \geq 0$,$f(x) = 7x + 8$. As $x \to 0^+$,$f(x) \to 8$.At $x = 0$,$f(0) = 8$.Since $f(x)$ is negative for $x 0$,$f(-h) = \frac{h}{2} > 0$,but $f(0) = 8$. However,we need $f(0) Statement-$I$ is false.For Statement-$II$: This is the standard definition of a local minimum at $x = a$. Thus,Statement-$II$ is true.

Class 12 Mathematics Applications of Derivatives Maxima and Minima

Medium

Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.
Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.

A
Statement-$I$ is true,Statement-$II$ is true; Statement-$II$ is a correct explanation for Statement-$I$.
B
Statement-$I$ is true,Statement-$II$ is true; Statement-$II$ is not a correct explanation for Statement-$I$.
C
Statement-$I$ is true,Statement-$II$ is false.
D
Statement-$I$ is false,Statement-$II$ is true.

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Applications of Derivatives

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Maxima and Minima

An extreme value of $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $(0, \pi / 2)$ is

MediumAP EAMCET 2023

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Number of solutions of the equation $3 \tan x + x^3 = 2$ in the interval $(0, \frac{\pi}{4})$ is:

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The height of a right circular cone of maximum volume inscribed in a sphere of diameter $a$ is-

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Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

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Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.

Similar Questions

An extreme value of $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $(0, \pi / 2)$ is

The maximum value of $y=x(\log x)^2$ is

Number of solutions of the equation $3 \tan x + x^3 = 2$ in the interval $(0, \frac{\pi}{4})$ is:

The height of a right circular cone of maximum volume inscribed in a sphere of diameter $a$ is-

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be such that $x = 0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$,then in the interval $[-1, 1]$:

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Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.
Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.