graph g(x)=4(x^3)-24x+9 on a calulator and estimate the local maxima and minima.
the answers are either
a)The local maximum is about –13.627417. The local minimum is about 31.627417.
b)The local maximum is about 31.627417. The local minimum is about –13.627417.
c)The local maximum is about 13.627417. The local minimum is about –31.627417.
d)The local maximum is about 22.627417. The local minimum is about –22.627417.
my calculator isn't working so i don't know what to do. Thanks for any help.
There is no need for a calculator.
The min or max are at x=+/-sqrt(2)
so I get
Maximum is 32.62742 (x=-(2^0.5)
Minimum is -13.6274 (x=2^0.5)
which is a)
But check my working.
I see. thank you!
y=-3x^2+9x-1
No problem! I can help you with that. Since your calculator isn't working, we'll have to estimate the local maxima and minima of the function manually.
To find the local maxima and minima of a function, we need to analyze the behavior of the function as it approaches critical points and changes direction.
The critical points occur when the derivative of the function is equal to zero or undefined. In this case, we need to find the derivative of the function.
The derivative of g(x) = 4x^3 - 24x + 9 can be found by differentiating each term. Since the derivative of a constant is zero, we only need to differentiate the first two terms:
g'(x) = (12x^2) - 24
Now, let's set the derivative equal to zero to find the critical points:
12x^2 - 24 = 0
Solving this equation, we get:
12x^2 = 24
x^2 = 2
x = ± √2
So, the critical points are x = √2 and x = -√2.
To estimate the local maxima and minima, we need to evaluate the function at these critical points as well as the points immediately adjacent to them. Since the function is continuous, we can estimate the local maximum and minimum by comparing the function values at these points.
Let's calculate the function values for these points:
g(-√2) ≈ 4(-√2)^3 - 24(-√2) + 9
≈ -13.627417
g(√2) ≈ 4(√2)^3 - 24(√2) + 9
≈ 31.627417
From the calculations, we can see that the local maximum is about 31.627417 and the local minimum is about -13.627417.
Therefore, the correct answer is option b) The local maximum is about 31.627417, and the local minimum is about -13.627417.