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Let $f(x)=4x^3-x^2-4x+2$ where $x\in[-1,1]$. Is it possible to find the maximum value of $f(x)$ without using calculus. Possibly through a series of inequality.

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    http://mathforum.org/library/drmath/view/62730.html , also, see the interesting: http://mathcircle.berkeley.edu/BMC4/Handouts/MaxMin.pdf – Moo Dec 31 '15 at 08:15
  • That comment is incredibly instructive, such translations of functions oft fall by the wayside – Triatticus Dec 31 '15 at 08:18
  • Here is another one: http://math.wvu.edu/~rhansen/Ryan_Hansen/Curriculum_Vitae_files/Mathematics%20Teacher%202008%20Hansen%20Optimization%20of%20cubic%20functions.pdf – Moo Dec 31 '15 at 08:21
  • @Variable you should post that as an answer – qwr Dec 31 '15 at 09:02

3 Answers3

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Thanks everyone for your answers but I was actually looking for an algebraic approach which I eventually figured out.

It suffice to observe that the given expression can be rewritten as $$\frac{1}{2} (1+2 x)^3-\frac{7}{4} (1+2 x)^2+\frac{13}{4}$$ Without the constant term, there is a double root at $x=-\frac{1}{2}$ which is a local maximum.

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You can approximate the extremum (i.e., maximum or minimum) in the interval $I := [-1,1]$ using numerical methods such as golden section search or other numerical methods. If you want to use golden section search, you should note that it only works with unimodal functions. So, one way to get around this condition is you can do the search on two different intervals. That is, apply the search on the interval $I_0 := [-1,0]$ and the search on the interval $I_1 := [0,1]$ separately since the function is unimodal on those intervals (according to its graph) and $I = I_0 \cup I_1$.

Vincent
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By using R programming, i.e by using simulation method, you can find it easily:

x  = seq(-1, 1, length = 10000)
fx = 4 * x^3 - x^2 -4*x +2
maxValue = max(fx)
# plot to visualize it
plot(x, fx)

you can check in the image as well

krishna Prasad
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