Questions: The function f(x)=2x^3-24x^2+42x+9 has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema. This function has a local minimum at x= with output value: and a local maximum at x= with output value:

The function f(x)=2x^3-24x^2+42x+9 has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema.

This function has a local minimum at x= with output value: and a local maximum at x= with output value:

Transcript text: The function $f(x)=2 x^{3}-24 x^{2}+42 x+9$ has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema. This function has a local minimum at $x=$ $\square$ with output value: $\square$ and a local maximum at $x=$ $\square$ with output value: $\square$

Solution

Solution Steps

Step 1: Find the derivative of the function

The derivative of the function $f(x) = 2x^3 - 24x^2 + 42x + 9$ is $f'(x) = 6_x^2 - 48_x + 42$.

Step 2: Solve for critical points

The critical points are found by solving $f'(x) = 0$, which gives $x$ values as: 1, 7.

Step 3: Determine the nature of critical points

At $x = 1$, the function has a local maximum with a value of $f(x) = 29$.

At $x = 7$, the function has a local minimum with a value of $f(x) = -187$.

Final Answer:

The function has a local maximum at $x = 1$ with a value of $f(x) = 29$. The function has a local minimum at $x = 7$ with a value of $f(x) = -187$.

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!