For the function f(x)=x^3-2x^2-4x-5, which of the following statement is not correct ? Option 1: The function f(x) has local minima at x=2 Option 2: The local maximum value of f(x) at x=-2/3 is 13/27. Option 3: The turning points of the given curve are (2,-13) and (-2/3,-95/27) Option 4: The function has local maxima at x=-2/3

Answer: The local maximum value of f(x) at x=-2/3 is 13/27. Solution: f'(x) = 3x^2-4x-4 = (3x+2)(x-2), so the critical points are x=-2/3 and x=2. The sign change gives a local maximum at x=-2/3 and a local minimum at x=2. f(2) = -13 and f(-2/3) = -95/27, not 13/27, so option 2 is the incorrect statement.

Calculus

Application of Derivatives

Medium

applied

For the function $f(x)=x^3-2x^2-4x-5$ , which of the following statement is not correct ?

The function $f(x)$ has local minima at $x=2$

The local maximum value of $f(x)$ at $x=-\dfrac{2}{3}$ is $\dfrac{13}{27}$ .

The turning points of the given curve are $(2,-13)$ and $\left(-\dfrac{2}{3},-\dfrac{95}{27}\right)$

The function has local maxima at $x=-\dfrac{2}{3}$

✅ Correct Option: 2

$f'(x) = 3x^2-4x-4 = (3x+2)(x-2)$ , so the critical points are $x=-\dfrac{2}{3}$ and $x=2$ .

The sign change gives a local maximum at $x=-\dfrac{2}{3}$ and a local minimum at $x=2$ .

$f(2) = -13$ and $f\left(-\dfrac{2}{3}\right) = -\dfrac{95}{27}$ , not $\dfrac{13}{27}$ , so option 2 is the incorrect statement.

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For the function f(x)=x3−2x2−4x−5f(x)=x^3-2x^2-4x-5f(x)=x3−2x2−4x−5, which of the following statement is not correct ?

Solution

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For the function $f(x)=x^3-2x^2-4x-5$ , which of the following statement is not correct ?