The question
A function f(x)=10−x−2x2 is increasing on the interval
(A) (−∞,−41]
(B) (−∞,41)
(C) [−41,∞)
(D) [−41,41]
Exam answer
f(x)=10−x−2x2, so f′(x)=−1−4x.
Why?
Differentiate term by term with dxdxn=nxn−1: the constant 10 gives 0, −x gives −1, and −2x2 gives −2×2x=−4x.
For an increasing function, f′(x)≥0:
Why?
If f′(x)≥0 on an interval, f is increasing there. Here f′(x)=0 only at the single point x=−41, so that end point can be included.
Why?
Multiplying an inequality by −1 reverses its sign, so ≥ becomes ≤.
Then subtract 1 to get 4x≤−1, and divide by 4, a positive number, which keeps the sign.
−1−4x1+4x4xx=−(1+4x)≤0≤0−1=−1≤4−1=−41
∴ x∈(−∞,−41]
Why?
Check with the graph: f is a parabola opening downwards (the x2 term is negative). Writing it as ax2+bx+c, a=−2 and b=−1, so its top is at
x=−2ab=−2(−2)−1=−41=−41
It rises up to the top and falls after it. So (C), to the right of −41, is where f decreases.
Answer: (A) (−∞,−41]
Where this comes fromChapter 6, Application of Derivatives: Increasing and decreasing functions.
Question type: Find where a function increases and decreases
How to spot it: click to see
Type 5. Find where a function increases and decreases
Spot it: "Find the intervals in which f is increasing or decreasing", or "strictly increasing or strictly decreasing".
Here: f′(x)=−1−4x is 0 at x=−41, which cuts the line in two. It is positive to the left, so f is increasing on (−∞,−41].
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