Question

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}} )

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 \(c\)

$$
LATEXBLOCK0
$$

Answer:

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