The function f(x) = (1/2)^x on R is | vedclass

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

Question

(A) Let $f(x) = (1/2)^x$ for all $x \in R$. To determine the nature of the function,we find its derivative: $f'(x) = \frac{d}{dx} \left( (1/2)^x \righ

Vedclass · Accepted Answer

Strictly decreasing

Vedclass · Answer

(A) Let $f(x) = (1/2)^x$ for all $x \in R$. To determine the nature of the function,we find its derivative: $f'(x) = \frac{d}{dx} \left( (1/2)^x \right) = (1/2)^x \ln(1/2)$. Since $\ln(1/2) = \ln(1) - \ln(2) = 0 - \ln(2) = -\ln(2)$,we have: $f'(x) = -(1/2)^x \ln(2)$. Since $(1/2)^x > 0$ for all $x \in R$ and $\ln(2) > 0$,it follows that $f'(x) Because the derivative $f'(x)$ is strictly less than $0$ for all $x$ in the domain,the function $f(x)$ is strictly decreasing on $R$.

The function $f(x) = (1/2)^x$ on $R$ is

Similar Questions

The function $f(x) = x^3 - 3x$ is....

If the tangent drawn to the curve $y=x^3-ax^2+x+1$ at each point $x \in R$ is inclined at an acute angle with the positive direction of the $X$-axis,then the set of all possible values of '$a$' is

Let $f(x) = \cos \left(\frac{\pi}{x}\right), x \neq 0$. Assuming $k$ is an integer,which of the following is true?

For what values of $x$ is the function $f(x) = x + \frac{1}{x}, x \neq 0$ strictly increasing?

Let $f: [0, 2] \to R$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in (0, 2)$. If $\phi(x) = f(x) + f(2 - x)$,then $\phi$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module