Finding all the set of values for k such that $-2x^3+24x+8=k$ find all the set of values for k such that $-2x^3+24x+8=k$ has more than one solution.
This problem can only be solved using basic Calculus 1 knowledge.
I thought about setting the equation to zero and then computing the x value that can then be rewritten in terms of k but I am not sure that is the correct approach can someone help me?
 A: Let $f(x) = -2x^3+24x+8 = k$
Now Let $f(x) = -2x^3+24x+8$ and we will find the nature of cure and plot the curve in $X-Y$ plane
So $f^{'}(x) = -6x^2+24$ and for Max. and Min. $f^{'}(x) = 0$
we get $-6(x^2-4) = 0 \Rightarrow x = \pm 2$
Now $f^{''}(x) = -12x$
So at $x = -2$ , we get $f^{''}(-2) = -12 \times -2 = 24>0$
So $x = -2$ is a point of Minimum
Similarly at $x = 2$ , we get $f^{''}(2) = -12 \times 2 = -24>0$
So $x = 2$ is a point of Maximum
So $f(-2) \leq f(x) \leq f(+2)$
So $-24 \leq f(x) \leq 40$
So value of $k$ for which the equation $f(x) = -2x^3+24x+8$ has more then one real roots is
$-24 < k < 40\Rightarrow k\in \left(-24,40\right)$
A: Hint: For the function to have more than one solution, it must have two distinct, real turning points. For what values of $k$ do those turning points occur on opposite sides of the $x$ axis?
(where we're taking the function to be $y=-2x^3+24x+8-k$)
A: Define $f:\mathbb{R} \rightarrow \mathbb{R}$ with $f(x)= -2x^3+24x+8$. Study this function with the regular first derivative stuff considerations. You will have no problems with this as the derivative of $f$ is a second degree polynomial and you can then calculate ir zeros explicitly. Now sketch a graph. For which values of $k \in \mathbb{R}$ does the line $y=k$ has, at least, two points of intersection with the graph of $f$?
Note: the last part may not be totally formal but it can be modified to make it fulfill formality requirements. One possibility is to go like this: break the real axis into three intervals such that $f$ is injective in each one and then count the number of solutions in each piece in a formal way(exploiting the injectivity of $f$)
