Let f be the function given by f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2. Which of the following statements is true?
f′(0.4)
We can use the properties of derivatives to determine the validity of the following statements:
1. The function f has no critical points.
We can find the critical points by setting f'(x) = 0:
f'(x) = x^6 (7x + 72 – 5x^2 – 15x + 8) = 0
This equation has six solutions, which are the critical points of f. Therefore, statement 1 is false.
2. The function f has a local maximum at x=0.
To determine if there is a local maximum at x=0, we can examine the sign of f'(x) near x=0. We have:
f'(x) = (1/7)x^6(7x+72-5x^2-15x+8)
For x slightly less than 0, f'(x) is negative, indicating a decreasing slope. For x slightly greater than 0, f'(x) is positive, indicating an increasing slope. Therefore, f has a local minimum at x=0. Statement 2 is false.
3. The function f is concave down on the interval (-infinity, infinity).
To determine the concavity of f, we compute the second derivative:
f”(x) = 12x^4(7x+12-5x^2-15x+8) + 2x^3(7-15x) + 2x(4)
The discriminant of the quadratic term in the expression above is -1911, which is negative. Therefore, the second derivative is negative for all x, implying that f is concave down on the entire real line. Statement 3 is true.
4. The function f has a relative maximum at x=1.
To determine whether x=1 is a relative maximum, we can examine the sign of f'(x) near x=1. We have:
f'(x) = (1/7)x^6(7x+72-5x^2-15x+8)
For x slightly less than 1, f'(x) is positive, indicating an increasing slope. For x slightly greater than 1, f'(x) is negative, indicating a decreasing slope. Therefore, x=1 is a relative maximum. Statement 4 is true.
In summary, only statements 3 and 4 are true.
More Answers:
Continuity Of Functions: How To Determine Existence Of F(2)
Find Equation Of Line Tangent To Graph Of F At X=-1 | Step-By-Step Guide
Calculating Average Rate Of Change Of A Math Function Over An Interval: Step-By-Step Guide
Error 403 The request cannot be completed because you have exceeded your quota. : quotaExceeded
We can use the properties of derivatives to determine the validity of the following statements:
1. The function f has no critical points.
We can find the critical points by setting f'(x) = 0:
f'(x) = x^6 (7x + 72 – 5x^2 – 15x + 8) = 0
This equation has six solutions, which are the critical points of f. Therefore, statement 1 is false.
2. The function f has a local maximum at x=0.
To determine if there is a local maximum at x=0, we can examine the sign of f'(x) near x=0. We have:
f'(x) = (1/7)x^6(7x+72-5x^2-15x+8)
For x slightly less than 0, f'(x) is negative, indicating a decreasing slope. For x slightly greater than 0, f'(x) is positive, indicating an increasing slope. Therefore, f has a local minimum at x=0. Statement 2 is false.
3. The function f is concave down on the interval (-infinity, infinity).
To determine the concavity of f, we compute the second derivative:
f”(x) = 12x^4(7x+12-5x^2-15x+8) + 2x^3(7-15x) + 2x(4)
The discriminant of the quadratic term in the expression above is -1911, which is negative. Therefore, the second derivative is negative for all x, implying that f is concave down on the entire real line. Statement 3 is true.
4. The function f has a relative maximum at x=1.
To determine whether x=1 is a relative maximum, we can examine the sign of f'(x) near x=1. We have:
f'(x) = (1/7)x^6(7x+72-5x^2-15x+8)
For x slightly less than 1, f'(x) is positive, indicating an increasing slope. For x slightly greater than 1, f'(x) is negative, indicating a decreasing slope. Therefore, x=1 is a relative maximum. Statement 4 is true.
In summary, only statements 3 and 4 are true.
More Answers:
Continuity Of Functions: How To Determine Existence Of F(2)Find Equation Of Line Tangent To Graph Of F At X=-1 | Step-By-Step Guide
Calculating Average Rate Of Change Of A Math Function Over An Interval: Step-By-Step Guide