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.\nf(x)=6x^{2}-5x - 1\nthe relative extreme point is \n(type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression.)

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.\nf(x)=6x^{2}-5x - 1\nthe relative extreme point is \n(type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Find the derivative

The derivative of $f(x)=6x^{2}-5x - 1$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=12x-5$.

Step2: Set the derivative equal to zero

Set $f'(x) = 0$, so $12x-5=0$. Solving for $x$ gives $x=\frac{5}{12}$.

Step3: Find the second - derivative

The second - derivative $f''(x)$ of $f'(x)=12x - 5$ is $f''(x)=12$. Since $f''(\frac{5}{12})=12>0$, the function has a relative minimum at $x = \frac{5}{12}$.

Step4: Find the $y$ - value

Substitute $x=\frac{5}{12}$ into the original function $f(x)=6x^{2}-5x - 1$. [ \begin{align*} f(\frac{5}{12})&=6\times(\frac{5}{12})^{2}-5\times\frac{5}{12}-1\ &=6\times\frac{25}{144}-\frac{25}{12}-1\ &=\frac{25}{24}-\frac{50}{24}-\frac{24}{24}\ &=\frac{25 - 50-24}{24}\ &=-\frac{49}{24} \end{align*} ]

Answer:

$(\frac{5}{12},-\frac{49}{24})$