let f be the function given by f(x) = 5x² - 3. find and simplify (f(x + h)-f(x))/h. a 5x + 10h b 5x - 10h c…

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x): [ \begin{align*} f(x + h)&=5(x + h)^2-3\ &=5(x^{2}+2xh+h^{2})-3\ &=5x^{2}+10xh + 5h^{2}-3 \end{align*} ]

Step2: Calculate f(x + h)-f(x)

[ \begin{align*} f(x + h)-f(x)&=(5x^{2}+10xh + 5h^{2}-3)-(5x^{2}-3)\ &=5x^{2}+10xh + 5h^{2}-3 - 5x^{2}+3\ &=10xh+5h^{2} \end{align*} ]

Step3: Calculate (\frac{f(x + h)-f(x)}{h})

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{10xh + 5h^{2}}{h}\ &=\frac{h(10x + 5h)}{h}\ &=10x+5h \end{align*} ]

Answer:

C. 10x + 5h