Derivatives and relative max's and min's

In summary, to find relative maximums, minimums, or neither for f(x)=x^3-12x^2+15x+16, you need to use the first derivative which is 3x^2-24x+15. Then, set it equal to 0 and solve for x to get the roots x=4+-sqrt(11). These are the correct x-values to use when plugging back into f(x) to find the corresponding y-values.
  • #1
kendalgenevieve
6
0
f(x)=x^3-12x^2+15x+16
Use the first derivative to find relative maximums, minimums, or neither.

I am trying to find x to plug it back into f(x) to get my y value, but I am not sure if I am getting the correct x value. I did the first derivative and got 3x^2-24x+15. I then set it equal to 0 and got x=4+- the square root of 44.
Is this correct or am I getting the wrong x-value?
 
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  • #2
You have correctly computed $f'$, however the roots of $f'$ are not quite correct. These roots are in fact:

\(\displaystyle x=4\pm\sqrt{11}\)
 

FAQ: Derivatives and relative max's and min's

What are derivatives?

Derivatives are a mathematical concept that measures the rate at which a dependent variable changes with respect to an independent variable. In other words, it calculates the slope or rate of change of a function at a specific point.

Why are derivatives important?

Derivatives are important in various fields such as physics, economics, and engineering as they help us understand and analyze how variables change over time. They also have practical applications in optimization and predicting future trends.

How do you find the derivative of a function?

The process of finding a derivative involves using rules such as the power rule, product rule, and chain rule. These rules allow us to find the derivative of a function by manipulating its algebraic expression. Additionally, we can also use graphing software or online calculators to find derivatives.

What are relative maxima and minima?

Relative maxima and minima are points on a graph where the function reaches its highest and lowest values, respectively, in a specific interval. They are also known as local maxima and minima as they are only applicable within a certain range and may not be the absolute maximum or minimum of the entire function.

How do you identify relative maxima and minima?

To identify relative maxima and minima, we can use the first derivative test or the second derivative test. The first derivative test involves finding the critical points of the function and determining whether they are relative maxima or minima by evaluating the sign of the derivative at those points. The second derivative test involves finding the concavity of the function at the critical points to determine whether they are points of inflection, maxima, or minima.

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