The given function is
[tex]f(x)=-10x^2-120x-5[/tex]
First, find the first derivative of the function f(x). Use the power rule.
[tex]\begin{gathered} f^{\prime}(x)=-10\cdot2x^{2-1}-120x^{1-1}+0 \\ f^{\prime}(x)=-20x-120 \end{gathered}[/tex]
Then, make it equal to zero.
[tex]-20x-120=0[/tex]
Solve for x.
[tex]\begin{gathered} -20x=120 \\ x=\frac{120}{-20} \\ x=-6 \end{gathered}[/tex]
This means the function has one critical value that creates two intervals.
We have to evaluate the function using two values for each interval.
Let's evaluate first for x = -7, which is inside the first interval.
[tex]f^{\prime}(-7)=-20(-7)-120=140-120=20\to+[/tex]
Now evaluate for x = -5, which is inside the second interval.
[tex]f^{\prime}(7)=-20(-5)-120=100-120=-20\to-[/tex]
As you can observe, the function is increasing in the first interval but decreases in the second interval. This means when x = -6, there's a maximum point.
At last, evaluate the function when x = -6 to find the y-coordinate and form the point.
[tex]\begin{gathered} f(-6)=-10(-6)^2-120(-6)-5=-10(36)+720-6 \\ f(-6)=-360+720-5=355 \end{gathered}[/tex]
Therefore, we have a relative maximum point at (-6, 355).
