find the percentage rate of change of f(x) at the indicated value of x. f(x)=9750 - 5x²; x = 25 the…

Explanation:

Step1: Find the derivative of f(x)

The derivative of $f(x)=9750 - 5x^{2}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ and $\frac{d}{dx}(C)=0$ (where $C$ is a constant) is $f^\prime(x)=\frac{d}{dx}(9750)-5\frac{d}{dx}(x^{2})=0 - 5\times2x=-10x$.

Step2: Evaluate f(x) and f'(x) at x = 25

First, find $f(25)$: $f(25)=9750-5\times(25)^{2}=9750 - 5\times625=9750 - 3125 = 6625$. Then, find $f^\prime(25)$: $f^\prime(25)=-10\times25=-250$.

Step3: Calculate the percentage rate of change

The formula for the percentage rate of change of a function $y = f(x)$ at $x=a$ is $\frac{f^\prime(a)}{f(a)}\times100%$. Substitute $a = 25$, $f(25)=6625$ and $f^\prime(25)=-250$ into the formula: $\frac{-250}{6625}\times100%=\frac{-25000}{6625}% \approx - 3.8%$.

Answer:

$-3.8$