Find the maximum value of the function [tex]f ( x )=x^3+3 x^2-9 x +1[/tex]
(a) 12
(b) 28
(c) -4
(d) 1

The maximum value of the function f ( x ) = x 3 + 3 x 2 − 9 x + 1 is found to be 28 at the critical point x = − 3 . This is determined through the first and second derivative tests. Therefore, the chosen option is (b) 28.
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Answered by Anonymous | 2025-07-05

Find the first derivative of the function f ( x ) = x 3 + 3 x 2 − 9 x + 1 , which is f ′ ( x ) = 3 x 2 + 6 x − 9 .
Set the first derivative to zero and solve for x to find the critical points: x = − 3 and x = 1 .
Use the second derivative test to determine that x = − 3 is a local maximum.
Evaluate the function at x = − 3 to find the maximum value: f ( − 3 ) = 28 . The maximum value is 28 ​ .

Explanation

Problem Analysis We are given the function f ( x ) = x 3 + 3 x 2 − 9 x + 1 and asked to find its maximum value.

Finding the First Derivative To find the maximum value, we first need to find the critical points of the function. We do this by taking the first derivative and setting it equal to zero. The first derivative is: f ′ ( x ) = 3 x 2 + 6 x − 9

Finding Critical Points Now, we set the first derivative equal to zero and solve for x :
3 x 2 + 6 x − 9 = 0 Divide by 3: x 2 + 2 x − 3 = 0 Factor the quadratic: ( x + 3 ) ( x − 1 ) = 0 So the critical points are x = − 3 and x = 1 .

Finding the Second Derivative To determine whether these critical points are local maxima or minima, we find the second derivative: f ′′ ( x ) = 6 x + 6

Determining Maxima and Minima Now, we evaluate the second derivative at the critical points: For x = − 3 :
f ′′ ( − 3 ) = 6 ( − 3 ) + 6 = − 18 + 6 = − 12 Since f ′′ ( − 3 ) < 0 , there is a local maximum at x = − 3 .
For x = 1 :
f ′′ ( 1 ) = 6 ( 1 ) + 6 = 12 Since 0"> f ′′ ( 1 ) > 0 , there is a local minimum at x = 1 .

Evaluating the Function at the Maximum Now we evaluate the function at the local maximum x = − 3 :
f ( − 3 ) = ( − 3 ) 3 + 3 ( − 3 ) 2 − 9 ( − 3 ) + 1 = − 27 + 27 + 27 + 1 = 28

Checking for Global Maximum Since the function is a cubic polynomial, it goes to ∞ as x goes to ∞ and to − ∞ as x goes to − ∞ . Therefore, the local maximum is not a global maximum. However, we are given a set of options, and 28 is among them.

Final Answer Therefore, the maximum value of the function is 28.


Examples
Understanding how to find maximum values of functions is crucial in many real-world applications. For example, in business, you might want to maximize profit, which can be modeled as a function of various factors like production cost and sales price. By finding the critical points and determining the maximum value, you can optimize your business strategy. Similarly, in physics, you might want to find the maximum height reached by a projectile, which involves finding the maximum value of a parabolic function.

Answered by GinnyAnswer | 2025-07-05