Let f- - be defined asfx- -55x- if x4Let A-x - - f is increasing- Then A is equal to

F'(x)={ nbsp; 55 amp;; nbsp; amp;x lt; 5 6(x^2 x 20) nbsp; amp;; nbsp; amp; 5≤ x ≤ 4 6(x^2 x 6) nbsp; amp;; nbsp; amp; x g ...

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Standard XII

Mathematics

Monotonically Increasing Functions

Let f: ℝ→ℝ be...

Question

Let $f : R \to R$ be defined as
$f (x) = ⎧ ⎨ ⎩ \begin{matrix} - 55 x, & if x < - 5 2 x^{3} - 3 x^{2} - 120 x, & if - 5 \leq x \leq 4 2 x^{3} - 3 x^{2} - 36 x - 336, & if x > 4 \end{matrix}$

Let $A = {x \in R : f$ is increasing $} .$ Then $A$ is equal to

(- 5, - 4) \cup (4, \infty)

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(- 5, \infty)

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(- \infty, - 5) \cup (4, \infty)

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(- \infty, - 5) \cup (- 4, \infty)

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Solution

The correct option is A $(- 5, - 4) \cup (4, \infty)$
$f^{'} (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} - 55 & ; & x < - 5 6 (x^{2} - x - 20) & ; & - 5 \leq x \leq 4 6 (x^{2} - x - 6) & ; & x > 4 \end{matrix}$

$\Rightarrow f^{'} (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} - 55 & ; & x < - 5 6 (x - 5) (x + 4) & ; & - 5 \leq x \leq 4 6 (x - 3) (x + 2) & ; & x > 4 \end{matrix}$

Hence, $f (x)$ is monotonically increasing in $(- 5, - 4) \cup (4, \infty)$

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Let f:R→R be defined as f(x)=⎧⎨⎩−55x,if x<−52x3−3x2−120x,if −5≤x≤42x3−3x2−36x−336,if x>4 Let A={x∈R:f is increasing}. Then A is equal to

The correct option is A (−5,−4)∪(4,∞)f′(x)=⎧⎪⎨⎪⎩−55;x<−56(x2−x−20);−5≤x≤46(x2−x−6);x>4 ⇒f′(x)=⎧⎪⎨⎪⎩−55;x<−56(x−5)(x+4);−5≤x≤46(x−3)(x+2);x>4 Hence, f(x) is monotonically increasing in (−5,−4)∪(4,∞)

Let $f : R \to R$ be defined as
$f (x) = ⎧ ⎨ ⎩ \begin{matrix} - 55 x, & if x < - 5 2 x^{3} - 3 x^{2} - 120 x, & if - 5 \leq x \leq 4 2 x^{3} - 3 x^{2} - 36 x - 336, & if x > 4 \end{matrix}$

Let $A = {x \in R : f$ is increasing $} .$ Then $A$ is equal to