the graph of the following function has one relative extreme point. find it and determine whether it is a…

the graph of the following function has one relative extreme point. find it and determine whether it is a relative maximum or a relative minimum. f(x)=8 + 2x - 5x^2. the relative extreme point is (1/5, 41/5) (type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression.) is the relative extreme point a relative maximum or a relative minimum? relative maximum relative minimum

the graph of the following function has one relative extreme point. find it and determine whether it is a relative maximum or a relative minimum. f(x)=8 + 2x - 5x^2. the relative extreme point is (1/5, 41/5) (type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression.) is the relative extreme point a relative maximum or a relative minimum? relative maximum relative minimum

Answer

Explanation:

Step1: Find the first - derivative

Given $f(x)=8 + 2x-5x^{2}$, using the power rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=2-10x$.

Step2: Set the first - derivative equal to zero

Set $f'(x) = 0$, so $2-10x=0$. Solving for $x$ gives $10x = 2$, then $x=\frac{1}{5}$.

Step3: Find the $y$ - value of the extreme point

Substitute $x = \frac{1}{5}$ into $f(x)$: $f(\frac{1}{5})=8+2\times\frac{1}{5}-5\times(\frac{1}{5})^{2}=8+\frac{2}{5}-\frac{1}{5}=\frac{40 + 2-1}{5}=\frac{41}{5}$. So the extreme point is $(\frac{1}{5},\frac{41}{5})$.

Step4: Find the second - derivative

Differentiate $f'(x)=2 - 10x$ with respect to $x$. Using the power rule, $f''(x)=-10$.

Step5: Determine if it's a maximum or minimum

Since $f''(\frac{1}{5})=-10<0$, the function has a relative maximum at the point $(\frac{1}{5},\frac{41}{5})$.

Answer:

Relative maximum