Find:
(a) the interval(s) on which f is increasing,
(b) the interval(s) on which f is decreasing,
(c) the open interval(s) on which f is concave up,
(d) the open interval(s) on which f is concave down, and
(e) the X-coordinate(s) of any/all inflection point(s).
f(x)= (x^4) - (4802 x^2) + 9604
take the first derivative ....
a) f is increasing for all values of x for which f '(x) is positive
f is decreasing for all values of x for which f '(x) is negative
take the second derivative
c) if the second derivative is positive, f is concave up
d) if the second derivative is negative, f is concave down
e) set the second derivative equal to zero and solve for x
f '(x) = 4x^3 - 9604x
= 4x(x^2 - 2401)
= 4x(x-49)(x+49)
What conclusions can you draw from that?
Use what you know about the properties and general shape of y = x^4 + .....
i still did not understand. if u don't mind could you please explain it again and also the question asks for the intervals and the points. could you give the intervals and points for the the questions above in the questions .. Thanks a bunch :).
To determine the intervals of increasing and decreasing, we need to find the derivative of the function f(x) and identify where it is positive (increasing) or negative (decreasing). To find the inflection points and intervals of concavity, we'll need to find the second derivative of f(x) and analyze its sign changes. Let's go step by step:
Step 1: Find the first derivative of f(x):
f'(x) = d/dx[(x^4) - (4802 x^2) + 9604]
To find the derivative, we'll use the power rule and chain rule if necessary.
f'(x) = 4x^3 - 9604x
Step 2: Determine the intervals of increasing and decreasing:
To find where f(x) is increasing, we need to determine when f'(x) > 0.
4x^3 - 9604x > 0
Factor out 4x: 4x(x^2 - 2401) > 0
Now, we have two factors: 4x and (x^2 - 2401). We need to consider the sign of each factor separately.
For 4x > 0, x > 0.
For (x^2 - 2401) > 0, we can factor it as (x - 49)(x + 49) > 0.
This inequality is true when x < -49 or x > 49.
By checking the signs of both factors:
- When x < -49: 4x < 0 and (x^2 - 2401) > 0, so the overall inequality is true.
- When -49 < x < 0: 4x < 0, but (x^2 - 2401) < 0, so the overall inequality is false.
- When 0 < x < 49: 4x > 0, but (x^2 - 2401) < 0, so the overall inequality is false.
- When x > 49: 4x > 0 and (x^2 - 2401) > 0, so the overall inequality is true.
Therefore, the function f(x) is increasing on the interval (-∞, -49) ∪ (0, 49).
To find where f(x) is decreasing, we need to determine when f'(x) < 0.
4x^3 - 9604x < 0
Factor out 4x: 4x(x^2 - 2401) < 0
Now, using the same factors, consider the signs:
- When x < -49: 4x < 0 and (x^2 - 2401) > 0, so the overall inequality is false.
- When -49 < x < 0: 4x < 0 and (x^2 - 2401) < 0, so the overall inequality is true.
- When 0 < x < 49: 4x > 0 and (x^2 - 2401) < 0, so the overall inequality is false.
- When x > 49: 4x > 0 and (x^2 - 2401) > 0, so the overall inequality is false.
Therefore, the function f(x) is decreasing on the interval (-49, 0).
Step 3: Find the second derivative of f(x):
f''(x) = d/dx[4x^3 - 9604x]
f''(x) = 12x^2 - 9604
Step 4: Determine the intervals of concavity:
To find where f(x) is concave up, we need to determine when f''(x) > 0.
12x^2 - 9604 > 0
Divide through by 4: 3x^2 - 2401 > 0
Factor: (x - 49)(x + 49) > 0
Again, analyze the signs:
- When x < -49: (x - 49) < 0 and (x + 49) < 0, so the overall inequality is true.
- When -49 < x < 49: (x - 49) < 0 and (x + 49) > 0, so the overall inequality is false.
- When x > 49: (x - 49) > 0 and (x + 49) > 0, so the overall inequality is true.
Therefore, the function f(x) is concave up on the interval (-∞, -49) ∪ (49, ∞).
To find where f(x) is concave down, we need to determine when f''(x) < 0.
12x^2 - 9604 < 0
Divide through by 4: 3x^2 - 2401 < 0
Factor: (x - 49)(x + 49) < 0
Analyze the signs again:
- When x < -49: (x - 49) < 0 and (x + 49) < 0, so the overall inequality is false.
- When -49 < x < 49: (x - 49) < 0 and (x + 49) > 0, so the overall inequality is true.
- When x > 49: (x - 49) > 0 and (x + 49) > 0, so the overall inequality is false.
Therefore, the function f(x) is concave down on the interval (-49, 49).
Step 5: Find the x-coordinates of any inflection points:
To find the inflection points, we need to locate where the concavity changes. In this case, f(x) changes from concave down to concave up at x = -49, and from concave up to concave down at x = 49.
The x-coordinate(s) of the inflection point(s) is/are -49 and 49.
To summarize:
(a) The function f(x) is increasing on the intervals (-∞, -49) ∪ (0, 49).
(b) The function f(x) is decreasing on the interval (-49, 0).
(c) The function f(x) is concave up on the intervals (-∞, -49) ∪ (49, ∞).
(d) The function f(x) is concave down on the interval (-49, 49).
(e) The x-coordinate(s) of the inflection point(s) is/are -49 and 49.