What is the minimum and maximum values respectively of the function $f(x) = 5x^3 – 2x^2 + a$ on the interval $(-2,2)$.
The answer among choices is $(-47,33)$,
though there are other choices like
$(-4,4)$, $(0.95,1)$, and $(0, 0.27)$.
I assume "a" is constant. So i try getting derivative and summing it to zero gives me:
$$
\frac{d}{dx} 5x^3 -2x^2 + a = x(15x-4) \Rightarrow 0 = x(15x-4)
$$
One root is zero, and the other is $15/4$. Is the problem set in some way wrong or am I missing something?
Best Answer
The maximum and minimum will depend on $a$, but their difference will not. You know the max and min come either at the roots of the derivative or at the ends of the interval. Note that $15/4$ is outside your interval, but $4/15$ is inside. So set $a=0$ as it will not matter for this and evaluate the function at those three points. The difference between the max and min should correspond to one of your choices. If you want, you can find the value of $a$ that makes the max and min be correct.