What are the coordinates of the inflection point of the cubic curve f(x) = x^3 - 6x^2 + 9x + 2?
Answer and explanation
Correct answer: C. (2, 4)
- Step 1: Find first derivative: f'(x) = 3x^2 - 12x + 9.
- Step 2: Find second derivative: f''(x) = 6x - 12.
- Step 3: Set f''(x) = 0 for concavity change: 6x - 12 = 0 => x = 2.
- Step 4: Evaluate f(2): f(2) = (2)^3 - 6(2)^2 + 9(2) + 2 = 8 - 24 + 18 + 2 = 4.
- Step 5: Verify sign change: f''(1) = -6 < 0 (concave down), f''(3) = 6 > 0 (concave up). Thus (2, 4) is the inflection point.
Answer
(2, 4).
Why not the others: '(1, 6)' is a local maximum point; '(3, 2)' is a local minimum point; '(0, 2)' is the y-intercept.
Practise more Calculus and Analytical Geometry.
Free questions with an explanation for every one, and full timed Qatar Senior Secondary Certificate (Thanaweya) mock tests.
More Calculus and Analytical Geometry questions
- Evaluate the limit: lim_(x -> 0) [sin(5x) / (2x)].
- Find the derivative dy/dx of the implicit equation x^3 + y^3 = 6xy at the point (3, 3) on the Folium of Descartes.
- What is the equation of the normal line to the curve y = x^2 - 4x + 5 at the point (1, 2)?
- Evaluate the definite integral: integral from 0 to 1 of [x / (x^2 + 1)] dx.
- Find the absolute maximum value of the function f(x) = 2x^3 - 3x^2 - 12x + 1 on the closed interval [-2, 3].
- What is the total area of the finite region enclosed between the parabola y = 4 - x^2 and the line y = 0?