calculator allowed let ( f(x)=sqrt{x} ). if the rate of change of ( f ) at ( x = c ) is twice its rate of…

calculator allowed let ( f(x)=sqrt{x} ). if the rate of change of ( f ) at ( x = c ) is twice its rate of change at ( x = 1 ), then ( c = ) (a) ( \frac{1}{4} ) (b) 1 (c) 4 (d) ( \frac{1}{sqrt{2}} ) (e) ( \frac{1}{2 sqrt{2}} )

calculator allowed let ( f(x)=sqrt{x} ). if the rate of change of ( f ) at ( x = c ) is twice its rate of change at ( x = 1 ), then ( c = ) (a) ( \frac{1}{4} ) (b) 1 (c) 4 (d) ( \frac{1}{sqrt{2}} ) (e) ( \frac{1}{2 sqrt{2}} )

Answer

Explanation:

Step1: Find the derivative of (f(x))

The function is (f(x)=\sqrt{x}=x^{\frac{1}{2}}). Using the power rule ((x^n)^\prime = nx^{n - 1}), we have (f^\prime(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}).

Step2: Evaluate the derivative at (x = c) and (x = 1)

  • The rate of change at (x = c) is (f^\prime(c)=\frac{1}{2\sqrt{c}}).
  • The rate of change at (x = 1) is (f^\prime(1)=\frac{1}{2\sqrt{1}}=\frac{1}{2}).

Step3: Set up the equation based on the given condition

Since the rate of change of (f) at (x = c) is twice its rate of change at (x = 1), we have (\frac{1}{2\sqrt{c}}=2\times\frac{1}{2}).

Step4: Solve the equation for (c)

[ \begin{align*} \frac{1}{2\sqrt{c}}&=1\ 2\sqrt{c}& = 1\ \sqrt{c}&=\frac{1}{2}\ c&=\frac{1}{4} \end{align*} ]

Answer:

A. (\frac{1}{4})