Assertion: Let f: R arrow R be a function such that f ( x )= x 3+ x 2+3 x + sin x. Then f is one-one. Reason: f ( x ) neither increasing nor decreasing function.

Question

Assertion: Let f: R arrow R be a function such that f ( x )= x 3+ x 2+3 x + sin x. Then f is one-one. Reason: f ( x ) neither increasing nor decreasing function. (A) Assertion is correct, reason is correct; reason is a correct explanation for assertion. (B) Assertion is correct, reason is correct; reason is not a correct explanation for assertion (C) Assertion is correct, reason is incorrect (D) Assertion is incorrect, reason is correct.

Tardigrade Admin · Accepted Answer

Every increasing or decreasing function is one-one

f^{'} (x) = 3 x^{2} + 2 x + 3 + cos x

= 3 (x + \frac{1}{3})^{2} + \frac{8}{3} + cos x > 0

[∵ ∣ cos x ∣ < 1 and 3 (x + \frac{1}{3})^{2} + \frac{8}{3} \geq \frac{8}{2}]

∴ f (x)

is strictly increasing

Q. Assertion : Let f:R→R be a function such that f(x)=x3+x2+3x+sinx. Then f is one-one. Reason : f(x) neither increasing nor decreasing function.

Assertion is correct, reason is correct; reason is a correct explanation for assertion.

Assertion is correct, reason is correct; reason is not a correct explanation for assertion

Assertion is correct, reason is incorrect

Assertion is incorrect, reason is correct.

Every increasing or decreasing function is one-one f′(x)=3x2+2x+3+cosx =3(x+31​)2+38​+cosx>0 [∵∣cosx∣<1and3(x+31​)2+38​≥28​] ∴f(x) is strictly increasing

Q. Assertion : Let $f : R \to R$ be a function such that $f (x) = x^{3} + x^{2} + 3 x + sin x$ . Then $f$ is one-one.
Reason : $f (x)$ neither increasing nor decreasing function.

Every increasing or decreasing function is one-one
$f^{'} (x) = 3 x^{2} + 2 x + 3 + cos x$
$= 3 (x + \frac{1}{3})^{2} + \frac{8}{3} + cos x > 0$
$[∵ ∣ cos x ∣ < 1 and 3 (x + \frac{1}{3})^{2} + \frac{8}{3} \geq \frac{8}{2}]$
$∴ f (x)$ is strictly increasing