"The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ is :"

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"$f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ ..... (i)

$f(x) = 3{x^2} - 6x - {3 \over 2}f''(2)$ ..... (ii)

$f''(x) = 6x - 6$ ..... (iii)

Now, is 3^rd equation

$f''(2) = 12 - 6 = 6$

$f''(11 = 0)$

Use (ii)

$f'(x) = 3{x^2} - 6x - {3 \over 2}f''(2)$

$f'(x) = 3{x^2} - 6x - {3 \over 2} \times 6$

$f'(x) = 3{x^2} - 6x - 9$

$f'(x) = 0$

$3{x^2} - 6x - 9 = 0$

$\Rightarrow$ x = $-$1 & 3

Use (iii)

$f''(x) = 6x - 6$

$f''( - 1) = - 12 < 0$ maxima

$f''(3) = 12 > 0$ minima.

Use (i)

$f(x) = {x^3} - 3{x^2} - {3 \over 2}f''(2)x + f''(1)$

$f(x) = {x^3} - 3{x^2} - {3 \over 2} \times 6 \times x + 0$

$f(x) = {x^3} - 3{x^2} - 9x$

$f(3) = 27 - 27 - 9 \times 3 = - 27$"

The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ is :

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The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ is :

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