Let f: R arrow R be defined as f(x)=(x3/3)+(x2/2)+a x+b. The least value of ' a ' for which f(x) is injective function, is

Question

Let f: R arrow R be defined as f(x)=(x3/3)+(x2/2)+a x+b. The least value of ' a ' for which f(x) is injective function, is (A)  (1/4) (B) (1/8) (C) (1/2) (D) 1

Tardigrade Admin · Accepted Answer

If f ( x ) is one-one then f ( x ) must be monotonic. Now, f prime(x)=x2+x+a ≥ 0 ∀ x ∈ R ⇒ D ≤ 0 i.c. 1-4 a ≤ 0 ⇒ a ≥ (1/4) So, a min =(1/4).

Q. Let $f : R \to R$ be defined as $f (x) = \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + a x + b$ . The least value of ' $a$ ' for which $f (x)$ is injective function, is

$\frac{1}{4}$

$\frac{1}{8}$

$\frac{1}{2}$

1

If $f (x)$ is one-one then $f (x)$ must be monotonic.
Now, $f^{'} (x) = x^{2} + x + a \geq 0\forall x \in R$
$\Rightarrow D \leq 0$ i.c. $1 - 4 a \leq 0 \Rightarrow a \geq \frac{1}{4}$
So, $a_{m i n} = \frac{1}{4}$ .

Q. Let f:R→R be defined as f(x)=3x3​+2x2​+ax+b. The least value of ' a ' for which f(x) is injective function, is

41​

81​

21​

1

If f(x) is one-one then f(x) must be monotonic. Now, f′(x)=x2+x+a≥0∀x∈R ⇒D≤0 i.c. 1−4a≤0⇒a≥41​ So, amin​=41​.

Q. Let $f : R \to R$ be defined as $f (x) = \frac{x ^{3}}{3} + \frac{x ^{2}}{2} + a x + b$ . The least value of ' $a$ ' for which $f (x)$ is injective function, is

$\frac{1}{4}$

$\frac{1}{8}$

$\frac{1}{2}$

If $f (x)$ is one-one then $f (x)$ must be monotonic.
Now, $f^{'} (x) = x^{2} + x + a \geq 0\forall x \in R$
$\Rightarrow D \leq 0$ i.c. $1 - 4 a \leq 0 \Rightarrow a \geq \frac{1}{4}$
So, $a_{m i n} = \frac{1}{4}$ .