Which is true for the graph of f(x)= -x^3+3x-2?
Answer: relative maximum at (1,0)
2) Determine between which consecutive integers one or more real zeros of f(x)= 3x^4+x^3-2x^2+4 are located.
Answer: -1 and 0
To determine the relative maximum of the function f(x)= -x^3+3x-2, we need to find the critical points of the function and determine their nature. Here's how we can do it step by step:
1) Find the derivative of the function f(x) with respect to x. The derivative of f(x) will help us locate the critical points where the slopes of the function change.
f'(x) = -3x^2 + 3
2) Set the derivative equal to zero and solve for x to find the critical points:
-3x^2 + 3 = 0
-3x^2 = -3
x^2 = 1
x = ±1
3) Plug the critical points obtained in step 2 back into the original function f(x) to determine their nature as maximum, minimum, or neither:
For x = -1:
f(-1) = -(-1)^3+3(-1)-2 = -1+(-3)-2 = -6
So, the point (-1, -6) is a relative minimum.
For x = 1:
f(1) = -(1)^3+3(1)-2 = -1+3-2 = 0
So, the point (1, 0) is a relative maximum.
Therefore, the statement that is true for the graph of f(x) = -x^3+3x-2 is that it has a relative maximum at the point (1, 0).
Now let's move on to the second question.
To determine between which consecutive integers one or more real zeros of the function f(x) = 3x^4+x^3-2x^2+4 are located, we can use the Intermediate Value Theorem and test the function at different points.
1) Choose consecutive integers to test. In this case, we can test the function at -1 and 0.
2) Plug in -1 into the function f(x):
f(-1) = 3(-1)^4 + (-1)^3 - 2(-1)^2 + 4 = 3 + (-1) - 2 + 4 = 4
3) Plug in 0 into the function f(x):
f(0) = 3(0)^4 + 0^3 - 2(0)^2 + 4 = 4
Since f(-1) = 4 and f(0) = 4, we can conclude that there is at least one real zero of the function f(x) = 3x^4+x^3-2x^2+4 between the consecutive integers -1 and 0.
Therefore, the integer interval between which one or more real zeros of the function f(x) = 3x^4+x^3-2x^2+4 are located is -1 and 0.