The sum of all the local minimum values of the twice differentiable function f : R [] R de…

Question

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 :

Accepted Answer

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

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

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

Now, is 3rd 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)

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

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

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

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

Solutions · 1