The points of local maxima and local minima of the f(x) = x^3 - 6x^2 + 9x + 27 are Option 1: local maxima at x = 1 & local minima at x = 3 Option 2: local maxima at x = 3 & local minima at x = 1 Option 3: local maxima at x = -1 & local minima at x = -3 Option 4: local maxima at x = -3 & local minima at x = -1

Answer: local maxima at x = 1 & local minima at x = 3. Solution: f'(x)=3x^2-12x+9=3(x-1)(x-3). The sign of f' is positive, then negative, then positive, so f' changes +- at x=1 (local maximum) and -+ at x=3 (local minimum). Equivalently f''(x)=6x-12 is negative at x=1 and positive at x=3.

Calculus

Application of Derivatives

Easy

common

The points of local maxima and local minima of the $f(x) = x^3 - 6x^2 + 9x + 27$ are

local maxima at x = 1 & local minima at x = 3

local maxima at x = 3 & local minima at x = 1

local maxima at x = -1 & local minima at x = -3

local maxima at x = -3 & local minima at x = -1

✅ Correct Option: 1

$f'(x)=3x^2-12x+9=3(x-1)(x-3)$ . The sign of $f'$ is positive, then negative, then positive, so $f'$ changes $+\to-$ at $x=1$ (local maximum) and $-\to+$ at $x=3$ (local minimum). Equivalently $f''(x)=6x-12$ is negative at $x=1$ and positive at $x=3$ .

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D.	$f(1)$	IV.	Point of local maxima

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The points of local maxima and local minima of the f(x)=x3−6x2+9x+27f(x) = x^3 - 6x^2 + 9x + 27f(x)=x3−6x2+9x+27 are

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The points of local maxima and local minima of the $f(x) = x^3 - 6x^2 + 9x + 27$ are