The number of integral values of a for which | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) function $f(x)$ is invertible if and only if it is strictly monotonic (either strictly increasing or strictly decreasing) on its domain.For a cubi

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) function $f(x)$ is invertible if and only if it is strictly monotonic (either strictly increasing or strictly decreasing) on its domain.For a cubic polynomial $f(x)$,this occurs when $f'(x) \ge 0$ or $f'(x) \le 0$ for all $x \in R$.First,find the derivative: $f'(x) = 6x^2 - 6(a + 2)x + 12a$.Factor the derivative: $f'(x) = 6(x^2 - (a + 2)x + 2a) = 6(x - a)(x - 2)$.For $f(x)$ to be monotonic,$f'(x)$ must not change sign. However,since $f'(x)$ is a quadratic with two roots $a$ and $2$,it changes sign unless the roots are equal.Thus,we must have $a = 2$.If $a = 2$,$f'(x) = 6(x - 2)^2 \ge 0$ for all $x$,so the function is strictly increasing and thus invertible.Since $a = 2$ is the only integral value in the interval $[-4, 6]$ that satisfies the condition,the number of such values is $1$.

The number of integral values of $a$ for which the function $f: R \to R, f(x) = 2x^3 - 3(a + 2)x^2 + 12ax - 7$ where $a \in [-4, 6]$ is invertible,is

Similar Questions

If a function $f: R \rightarrow R$ is defined by $f(x) = \frac{4x}{5} + 3$,then $f^{-1}(x) =$

If the functions $f$ and $g$ are defined by $f(x) = 3x - 4$ and $g(x) = 2 + 3x$ for $x \in R$,then $g^{-1}(f^{-1}(5))$ is equal to

Let $A = \{1, 2, 3\}$ and $B = \{1, 3, 5\}$. $A$ relation $R: A \to B$ is defined by $R = \{(1, 3), (1, 5), (2, 1)\}$. Then ${R^{-1}}$ is defined by:

Let $f: A \rightarrow B$ and $g: B \rightarrow A$ be defined as $f(x)=x^2 \forall x \in A$ and $g(x)=x^{1/2} \forall x \in B$. $f(x)$ and $g(x)$ are inverse functions to each other when

Vedclass Test Series

Exam Paper Generator

Online Exam Module