Which of the following is true about f(x) = 3 | vedclass

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

Question

(C) Given the function $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$. Using the definitions $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \

Vedclass · Accepted Answer

$f$ is a strictly increasing function on $R$

Vedclass · Answer

(C) Given the function $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$. Using the definitions $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$: $f(x) = 3 \left( \frac{e^x - e^{-x}}{2} \right) - 2 \left( \frac{e^x + e^{-x}}{2} \right)$ $f(x) = \frac{3e^x - 3e^{-x} - 2e^x - 2e^{-x}}{2} = \frac{e^x - 5e^{-x}}{2}$ Now,find the derivative $f'(x)$: $f'(x) = \frac{d}{dx} \left( \frac{1}{2} e^x - \frac{5}{2} e^{-x} \right) = \frac{1}{2} e^x + \frac{5}{2} e^{-x}$ Since $e^x > 0$ and $e^{-x} > 0$ for all $x \in R$,it follows that $f'(x) > 0$ for all $x \in R$. Therefore,the function $f$ is a strictly increasing function on $R$. Hence,option $C$ is correct.

Which of the following is true about $f(x) = 3 \sinh(x) - 2 \cosh(x)$ for all $x \in R$?

Similar Questions

If $f(x) = 1 + x + \int_{1}^{x} (\ln^2 t + 2 \ln t) \, dt$,then $f(x)$ increases in

The function $f(x) = e^{-1/x}$ is strictly increasing for all $x$ where

If the function $f(x) = \cos |x| - 2ax + b$ is an increasing function on the entire real line,then what is the value of $a$?

For what values of $x$ is the function $f(x) = [x(x - 3)]^2$ an increasing function?

In which interval is the function $f(x) = 2x^2 - \ln |x|$ $(x \ne 0)$ monotonically decreasing?

Vedclass Test Series

Exam Paper Generator

Online Exam Module