Minimum Value
Find the minimum value of the f ( x ) = 9 x 2 − 6 x
The answer is -1.
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4 solutions
The problem should state that f ( x ) is a function with real domain, else there is no minimum. By the way, there's a comparatively shorter approach (known popularly as completing the square).
f ( x ) = 9 x 2 − 6 x = ( 3 x ) 2 − 2 ⋅ 3 x ⋅ 1 = ( 3 x − 1 ) 2 − 1
Making the implicit assumption that f ( x ) has real domain, i.e., f : R ↦ R , we have, by the trivial inequality , the minimum of f ( x ) as ( − 1 ) at x = 3 1 .
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Yes, although it is mostly understood, the question must mention that x ∈ R is the domain of f ( x ) .
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Yup! My point exactly! It's usually assumed implicitly that the function is on R for problems like this but it's better to have that given explicitly in the question so that there's no scope for ambiguity. :)
See, Karthik Venkata you gave me that book, and today I know differentiation!
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@Swapnil Das – Glad to know that the book was useful : ) .
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@Venkata Karthik Bandaru – I hope I could read that book everyday 😖
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@Swapnil Das – 😮 Aren't you able to read it everyday ?
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@Venkata Karthik Bandaru – No, other nonsense subjects come on my way! 😭
@Venkata Karthik Bandaru – Have you posted calculus problems?
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@Swapnil Das – Nope, I didn't.
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@Venkata Karthik Bandaru – Oh! Please post some!
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@Swapnil Das – I have now posted an NT problem. Check it out !
@Swapnil Das – What book is that ? :D
@Swapnil Das – Whats that book?:
Thank you for suggestion.
d x d ( 9 x 2 − 6 x ) = 1 8 x − 6
1 8 x − 6 = 0
x = 1 / 3
Plugging the value in the expression, f ( x ) = − 1
Mention the reason why did you differentiate the function and equate it with zero for the sake of beginners. :)
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The derivative of f(x) gives the slope of the function and by equating the derivative with zero it is possible to find the x-value for the function's minimum value (in the case of a parabola with a negative slope it would g give the max value).
We can use simple algebra. The function is a parabola with a positive coefficient, so the minimum lies at the vertex. Vertex has a value of f( -b / 2a ) where a=9 and b=6.
f ( x ) = 0 ⇔ x = 0 ∨ x = 3 2 V o r t e x → ( x v , y v ) x v = 2 0 + 2 / 3 = 1 / 3 y v = f ( 1 / 3 ) = − 1 ( m i n i m u m )
this question can be solved by basic algebra. we know that its is a quadratic function which has the leading coefficient(coefficient of x^2) positive so the minimum value of the function is (-D/4a) where 'D' id the discriminant(b^2-4ac) and 'a' is the leading coefficient.
hence the answer is -((6)^2-0)/4*9=-1