Let f(x) = x^3 + 8x + 3. Which one of the pro | vedclass

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

Question

(D) To determine if a function $f(x)$ has an inverse,we check if it is strictly monotonic (either strictly increasing or strictly decreasing).Given $f

Vedclass · Accepted Answer

$f'(x)$ is always positive.

Vedclass · Answer

(D) To determine if a function $f(x)$ has an inverse,we check if it is strictly monotonic (either strictly increasing or strictly decreasing).Given $f(x) = x^3 + 8x + 3$.The derivative is $f'(x) = \frac{d}{dx}(x^3 + 8x + 3) = 3x^2 + 8$.Since $x^2 \ge 0$ for all real $x$,it follows that $3x^2 \ge 0$.Therefore,$f'(x) = 3x^2 + 8 \ge 8$.Since $f'(x) > 0$ for all $x \in \mathbb{R}$,the function $f(x)$ is strictly increasing on its entire domain.$A$ strictly monotonic function is always one-to-one (injective) and thus possesses an inverse.Therefore,the property that $f'(x)$ is always positive allows us to conclude that $f(x)$ has an inverse.

Let $f(x) = x^3 + 8x + 3$. Which one of the properties of the derivative enables you to conclude that $f(x)$ has an inverse?

Similar Questions

$f: R \rightarrow R, f(x) = 3x + 2$ and $g: R \rightarrow R, g(x) = 6x + 5$. Find the value of $(g \circ f^{-1})(10)$.

Let $f$ be a real-valued function defined on the interval $(-1, 1)$ such that $e^{-x} f(x) = 2 + \int_0^x \sqrt{t^4 + 1} \, dt$,for all $x \in (-1, 1)$ and let $f^{-1}$ be the inverse function of $f$. Then $(f^{-1})'(2)$ is equal to

If $f(x) = x^{11} + \sin^3(35x) + 111x$,then the value of $f^{-1}(\sin \frac{\pi}{5}) + f^{-1}(\sin \frac{6\pi}{5}) + f^{-1}(\sin \frac{\pi}{7}) + f^{-1}(\sin \frac{8\pi}{7})$ is equal to

Consider $f: R_{+} \rightarrow [-5, \infty)$ given by $f(x) = 9x^{2} + 6x - 5$. Show that $f$ is invertible with $f^{-1}(y) = \frac{\sqrt{y+6}-1}{3}$.

Let $f(x) = (x - 1)^2 + 1$ for $x \ge 1$.
Statement-$1$: $S = \{x : f(x) = f^{-1}(x)\} = \{1, 2\}$.
Statement-$2$: $f$ is a bijection and $f^{-1}(x) = 1 + \sqrt{x - 1}$ for $x \ge 1$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module