F- given by fx-x3-3x2-12x-2sin x- then fx is

F(x)=x^3+3x^2+12x 2sin x ⇒ f'(x)=3x^2+6x+12 2cos x=3(x^2+2x+4) 2cos x =3[(x+1)^2+3] 2cos x gt; 0 ⇒ f'(x) gt;0 ∀ x∈ℝ So, nbsp;f(x) is increasing function ...

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Standard XII

Mathematics

Differentiability in an Interval

f:ℝ→ℝ given b...

Question

$f : R \to R$ given by $f (x) = x^{3} + 3 x^{2} + 12 x - 2 sin x,$ then $f (x)$ is

One-one function

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Many-one function

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Constant function

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None of these

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Solution

The correct option is A One-one function
$f (x) = x^{3} + 3 x^{2} + 12 x - 2 sin x$
$\Rightarrow f^{'} (x) = 3 x^{2} + 6 x + 12 - 2 cos x = 3 (x^{2} + 2 x + 4) - 2 cos x$
$= 3 [(x + 1)^{2} + 3] - 2 cos x > 0$
$\Rightarrow f^{'} (x) > 0 \forall x \in R$
So, $f (x)$ is increasing function and it is one-one function.

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Similar questions

Q. If

f (x) = x^{3} + 3 x^{2} + 12 x - 2 sin x

such that

f : R \to R

then

Q. If

f (x) = x^{3} + 3 x^{2} + 12 x - 2 sin x

where

f : R \to R

, then prove that

f

is bijective

Q. Show that the function

f

given by

f (x) = x^{3} - 3 x^{2} + 4 x, x \in R

is strictly increasing.

Q. Find the intervals in which the following functions are increasing or decreasing.
(i) f(x) = 10 − 6x − 2x²

(ii) f(x) = x² + 2x − 5

(iii) f(x) = 6 − 9x − x²

(iv) f(x) = 2x³ − 12x² + 18x + 15

(v) f(x) = 5 + 36x + 3x² − 2x³

(vi) f(x) = 8 + 36x + 3x² − 2x³

(vii) f(x) = 5x³ − 15x² − 120x + 3

(viii) f(x) = x³ − 6x² − 36x + 2

(ix) f(x) = 2x³ − 15x² + 36x + 1

(x) f(x) = 2x³ + 9x² + 12x + 20

(xi) f(x) = 2x³ − 9x² + 12x − 5

(xii) f(x) = 6 + 12x + 3x² − 2x³

(xiii) f(x) = 2x³ − 24x + 107

(xiv) f(x) = −2x³ − 9x² − 12x + 1

(xv) f(x) = (x − 1) (x − 2)²

(xvi) f(x) = x³ − 12x² + 36x + 17

(xvii) f(x) = 2x³ − 24x + 7

(xviii)

f (x) = \frac{3}{10} x^{4} - \frac{4}{5} x^{3} - 3 x^{2} + \frac{36}{5} x + 11

(xix) f(x) = x⁴ − 4x

(xx)

f (x) = \frac{x^{4}}{4} + \frac{2}{3} x^{3} - \frac{5}{2} x^{2} - 6 x + 7

(xxi) f(x) = x⁴ − 4x³ + 4x² + 15

(xxii) f(x) = 5x³^/2 − 3x⁵^/2, x > 0

(xxiii) f(x) = x⁸ + 6x²

(xxiv) f(x) = x³ − 6x² + 9x + 15

(xxv)

f (x) = {\{x (x - 2)\}}^{2}

Q. Prove that the function given by

f (x) = x^{3} - 3 x^{2} + 3 x - 100

is increasing on R.

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Differentiability in an Interval

Standard XII Mathematics

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f:R→R given by f(x)=x3+3x2+12x−2sinx, then f(x) is

The correct option is A One-one functionf(x)=x3+3x2+12x−2sinx ⇒f′(x)=3x2+6x+12−2cosx=3(x2+2x+4)−2cosx =3[(x+1)2+3]−2cosx>0 ⇒f′(x)>0 ∀ x∈R So, f(x) is increasing function and it is one-one function.

$f : R \to R$ given by $f (x) = x^{3} + 3 x^{2} + 12 x - 2 sin x,$ then $f (x)$ is