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The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists of five line segments as shown below. Given f (0) = 7, find the absolute minimum value of f (x) over the interval [–3, 0].

0
2.5
4.5
11.5

The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists-example-1
User ChristophK
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1 Answer

7 votes


f'(x)\ge0 for all
x in [-3, 0], so
f(x) is non-decreasing over this interval, and in particular we know right away that its minimum value must occur at
x=-3.

From the plot, it's clear that on [-3, 0] we have
f'(x)=-x. So


f(x)=\displaystyle\int(-x)\,\mathrm dx=-\frac{x^2}2+C

for some constant
C. Given that
f(0)=7, we find that


7=-\frac{0^2}2+C\implies C=7

so that on [-3, 0] we have


f(x)=-\frac{x^2}2+7

and


f(-3)=\frac52=2.5

User Amin Saadati
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