Consider the function f(x)=12x^5+60x^4−100x^3+4. For this function there are four important intervals: (−INF,A], [A,B] ,[B,C] , and [C,INF) where A, B, and C are the critical numbers. Find A, B, and C.
At each critical number A, B, and C does f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER.
For B I have 0 and for C I get 1
I guess A will be local max B will be neither, c local min
I don't know which will be the value of A because when I graph it, it's actually a value really close to 0, and I have pluged, -1, -0.1, -0.2, -0.3 and -0.5 but none of them are the answer.
Oh ok, I got the answer is -5, thanks anyways....
To find the critical numbers of the function f(x)=12x^5+60x^4−100x^3+4, we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
Step 1: Calculate the derivative of f(x). Let's call it f'(x).
f'(x) = 60x^4 + 240x^3 - 300x^2
Step 2: Set f'(x) equal to zero and solve for x to find the critical numbers.
60x^4 + 240x^3 - 300x^2 = 0
Step 3: Factor out x^2 from the equation.
x^2(60x^2 + 240x - 300) = 0
Step 4: Solve the quadratic equation 60x^2 + 240x - 300 = 0 either by factoring or using the quadratic formula.
When you solve this equation, you will find two values of x, which we'll call A and B.
Step 5: Check for any values of x where the derivative f'(x) is undefined. However, in this case, we only have polynomial terms in f(x), so it is defined for all real numbers.
Therefore, A and B are the critical numbers for the function f(x).
To determine whether f(x) has a local min, a local max, or neither at each critical number, we can use the second derivative test.
Step 6: Calculate the second derivative of f(x). Let's call it f''(x).
f''(x) = 240x^3 + 720x^2 - 600x
Step 7: Substitute the critical numbers A and B into f''(x) to evaluate the nature of the critical points.
- Substitute A into f''(x). If f''(A) > 0, then it's a local minimum. If f''(A) < 0, then it's a local maximum. If f''(A) = 0, then the test is inconclusive.
- Substitute B into f''(x) and follow the same steps.
Regarding the value of A that you mentioned, it seems that you have already tried different values near 0. Sometimes, finding the exact value can be challenging through numerical methods or graphing alone. In such cases, you may need to use an algebraic approach or more advanced methods like optimization algorithms to find the precise value.
Once you have determined the nature of each critical number (i.e., local min, local max, or neither), you can provide your answers as "LMIN," "LMAX," or "NEITHER" for each critical number.