Find all critical points of the function $\newline$ $f(x)=9x+|8x+1|$ $\newline$ (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) $\newline$ critical points:

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Q. Find all critical points of the function

\newline

f(x)=9x+|8x+1|

\newline

(Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.)

\newline

critical points:

Define critical points: Understand the definition of critical points. Critical points occur where the derivative of the function is zero or undefined. We need to find the derivative of $f(x) = 9x + |8x + 1|$ and solve for $x$ where the derivative is zero or does not exist.
Calculate derivative: Calculate the derivative of $f(x)$ . The absolute value function $|8x + 1|$ is piecewise, so we need to consider two cases: when $8x + 1 \geq 0$ and when $8x + 1 < 0$ . $\newline$ Case $1$ : When $8x + 1 \geq 0$ , $|8x + 1| = 8x + 1$ , and the derivative $f'(x) = \frac{d}{dx} (9x + 8x + 1) = 17$ . $\newline$ Case $2$ : When $8x + 1 < 0$ , $|8x + 1| = -(8x + 1)$ , and the derivative $f'(x) = \frac{d}{dx} (9x - 8x - 1) = 1$ .
Find derivative roots: Solve for $x$ where the derivative is zero. In Case $1$ , the derivative $f'(x) = 17$ is never zero. In Case $2$ , the derivative $f'(x) = 1$ is also never zero. Therefore, there are no critical points where the derivative is zero.
Identify undefined points: Determine if there are any points where the derivative is undefined. The derivative is undefined at the point where the piecewise function changes from one case to the other, which is at $8x + 1 = 0$ . Solve for $x$ : $8x + 1 = 0$ , $x = -\frac{1}{8}.$
Verify critical point: Verify that $x = -\frac{1}{8}$ is a critical point. Since the derivative changes from $17$ to $1$ at $x = -\frac{1}{8}$ , this is indeed a critical point because the derivative does not exist at this point (the function has a corner or cusp).