minimum value of f(x)

[as08998078]

(9) Let vector \(\displaystyle \, \overrightarrow{a}\, =\, (2,\, 3)\,\) and \(\displaystyle \, \overrightarrow{b}\, =\, (-1,\, 4).\,\) When \(\displaystyle \, \overrightarrow{x}\, -\, \overrightarrow{a}\, =\, \overrightarrow{b}\, -\, \overrightarrow{x},\,\) then \(\displaystyle \, \overrightarrow{x}\, =\, \boxed{\left(\dfrac{1}{2},\, \dfrac{7}{2}\right)}\)

(10) Let \(\displaystyle \, f(x)\, =\, -x^3\, +\, 6x^2\, -\, 9x\, +\, 1.\)

(i) The derivative \(\displaystyle \, f'(x)\, =\, \boxed{-3x^2\, +\, 12\, -\, 1}\)

(ii) Under \(\displaystyle \, 0\, \leq\, x\, \leq\, 3,\, \) the minimum value of \(\displaystyle \, f(x)\, \) is: _________

2: There is a parabola \(\displaystyle \, A\, :\, y\, =\, x^2\, -\, 4x\, -\, 5.\)

(1) The coordinate of the vertex of the parabola A is.....

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In (ii) I don't know how to find the minimum value of f(x). Please help me.
(In the box, it is answer)

 

[stapel]

(10) Let \(\displaystyle \, f(x)\, =\, -x^3\, +\, 6x^2\, -\, 9x\, +\, 1.\)

(i) The derivative \(\displaystyle \, f'(x)\, =\, \boxed{-3x^2\, +\, 12\, -\, 1}\)

(ii) Under \(\displaystyle \, 0\, \leq\, x\, \leq\, 3,\, \) the minimum value of \(\displaystyle \, f(x)\, \) is: _________

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In (ii) I don't know how to find the minimum value of f(x).

What have you learned about derivatives and their relation to max/min points? What do you know about finding the x-intercepts (that is, the zeroes) of quadratics?

 

[Deleted member 4993]

Since the domain is restricted, there could be global max/min at the end points.

 

[Deleted member 4993]

(9) Let vector \(\displaystyle \, \overrightarrow{a}\, =\, (2,\, 3)\,\) and \(\displaystyle \, \overrightarrow{b}\, =\, (-1,\, 4).\,\) When \(\displaystyle \, \overrightarrow{x}\, -\, \overrightarrow{a}\, =\, \overrightarrow{b}\, -\, \overrightarrow{x},\,\) then \(\displaystyle \, \overrightarrow{x}\, =\, \boxed{\left(\dfrac{1}{2},\, \dfrac{7}{2}\right)}\)

(10) Let \(\displaystyle \, f(x)\, =\, -x^3\, +\, 6x^2\, -\, 9x\, +\, 1.\)

(i) The derivative \(\displaystyle \, f'(x)\, =\, \boxed{-3x^2\, +\, 12\, -\, 1}\) ..... Incorrect \(\displaystyle \, f'(x)\, =\, {-3x^2\, +\, 12x\, -\, 9}\)

(ii) Under \(\displaystyle \, 0\, \leq\, x\, \leq\, 3,\, \) the minimum value of \(\displaystyle \, f(x)\, \) is: _________

2: There is a parabola \(\displaystyle \, A\, :\, y\, =\, x^2\, -\, 4x\, -\, 5.\)

(1) The coordinate of the vertex of the parabola A is.....

---

In (ii) I don't know how to find the minimum value of f(x). Please help me.
(In the box, it is answer)

.

 

[as08998078]

oh It's easy but why I don't know (may be i should learn more)
Thank you ^^