Looks easy but still you should try

Algebra Level 5

If $f(x) = 4x^3 - x^2 -2x+1$ and $g(x) = \begin{cases}{\text{min} [f(t) : 0 \le t \le x]} && {;\>0\le\>x\le\>1} \\ {3-x} && {;\>1\lt\>x\>\le\>2}\end{cases}$

Then find $g\left(\frac{1}{4}\right) + g\left(\frac{3}{4}\right) + g\left(\frac{5}{4}\right)$ .

The answer is 2.5.

1 solution

Purushottam Abhisheikh
Feb 26, 2015

$f(x)$ has a local minima and local maxima at $\dfrac{1}{2} and \dfrac{-1}{3}$ respectively. Lets sketch a rough diagram of $f(x)$ as below

Now we can define $g(x)$ as

$g(x) = \begin{cases}{f(x)} && {;\>0\>\le\>x\>\lt\>\frac{1}{2}} \\ {f(\frac{1}{2})} && {;\frac{1}{2}\>\le\>x\>\le\>1} \\ {3-x} && {;1\>\lt\>x\>\le\>2}\end{cases}$

So $g(\frac{1}{4}) = \frac{1}{2}$ , $g(\frac{3}{4}) = \frac{1}{4}$ and $g(\frac{5}{4}) = \frac{7}{4}$ .

Hence the required answer is $\frac{5}{2} = 2.5$

This should be a Calculus problem in my opinion. Other than that, it was a really nice problem.

Prasun Biswas - 6 years, 3 months ago

This question was given to me in a quiz of functions. So I added it in Algebra.

Purushottam Abhisheikh - 6 years, 3 months ago

Well, finding the extrema of higher degree polynomial functions is best done using derivates. That's why I suggested changing the topic. An algebraic method to find the extrema would be a lot more tedious for a cubic polynomial function anyway.

Prasun Biswas - 6 years, 3 months ago

@Prasun Biswas – Yeah you are right.

Purushottam Abhisheikh - 6 years, 3 months ago

@Prasun Biswas – @Prasun Biswas Could u please share the algebraic approach?

Abhinav Raichur - 5 years, 11 months ago

Looks easy but still you should try

The answer is 2.5.

1 solution

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