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A cubic function has the rule $y=f(x).$ The graph of the derivative function $f'$ crosses the $x$-axis at $(2,0)$ and $(-3,0).$ The maximum value of the derivative function is $10$.
The value of $x$ for which the graph of $y=f(x)$ has a local maximum is
$\eqalign{&{\bf A.} \ \ -2 \\ &{\bf B. } \ \ 2 \\ &{\bf C. } \ \ -3 \\ &{\bf D. } \ \ 3 \\ &{\bf E. } \ \ -\dfrac12 }$

What do I do with the information "maximum value of the derivative function is $10$? Does this mean the max value of the range of $f'(x) = 10$. How do I know where this point lies on the $f'(x)$ graph?

All I know is:

  • $f'(2) = 0$ which means its a stationary point for $f(x)$.
  • $f'(-3) = 0$ which means its another stationary point for $f(x)$.

Very confused. Can someone please help me interpret the question, and then solve for the local maximum.

The answer given is B:2

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4 Answers 4

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The answer is: $x = 2$.

$f'(x)$ has a graph that is a parabola opens down. This means that the graph of $f(x)$ increases from $-3$ to $2$, and then decreases.

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you have the roots of the derivative, and know it is a quadratic equation. Form the quadratic. Given that the derivative has a maximum, determine the sign of the $x^2$ term. From these, it should be possible to solve the question.

PS: I haven't verified if the answer you provided is correct.
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that $f'$ has a maximum implies that $f'$ is a quadratic polynomial with negative lead coefficient

therefore $f$ is a cubic polynomial with negative lead coeff (do you know how that looks like?)

it follows that there is a local minimum at $x=-3$ and a local maximum at $x=2$

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  • $\begingroup$ does it mean that f' has a maximum at 10, as in the turning point of the quadratic is at 10, therefore it is an upside down parabola? So if f' had a minimum at 10, it would a positive parabola - since f'(x) shape of a cubic is a quadratic graph.. $\endgroup$
    – confused
    Commented May 26, 2014 at 10:28
  • $\begingroup$ no, the maximum value is $10$, and since it is a parabola, it must open down $\endgroup$
    – Blah
    Commented May 26, 2014 at 11:43
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    $\begingroup$ like so: (open down) biology.arizona.edu/biomath/tutorials/quadratic/images/… - if so, then thats what i meant in the above ^ $\endgroup$
    – confused
    Commented May 26, 2014 at 12:20
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The derivative of a cubic function is a quadratic function. A quadratic function is bounded on one side, I mean it takes values in $[min,\ +\infty[$ or $]-\infty,\ max]$, depending on the sign of the leading term (2nd degree).

"The maximum value of the derivative function is 10" is information about this sign, and that will help you tell which roots of the derivative correspond to minima/maxima of the cubic.

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