5.2 Increasing and decreasing functionsIB Maths: Analysis and Approaches HL: Flashcards

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What does $f'(x)>0$ on an interval tell you about $f$?

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What does $f'(x)>0$ on an interval tell you about $f$ ?: $f$ is increasing on that interval.
What does $f'(x)<0$ on an interval tell you about $f$ ?: $f$ is decreasing on that interval.
What does $f'(a)=0$ mean graphically?: The tangent to $y=f(x)$ at $x=a$ is horizontal.
$f'(x)=3(x-2)(x+2)$ . Where is $f$ decreasing?: $-2<x<2$
$f'(x)=3(x-2)(x+2)$ . Where is $f$ increasing?: $x<-2$ or $x>2$
The graph of $y=f'(x)$ is below the $x$ -axis for $x<1$ . What does $f$ do there?: It is decreasing.
The graph of $y=f'(x)$ touches (but does not cross) the $x$ -axis at $x=a$ . What does $f$ do at $x=a$ ?: Its tangent is horizontal but it keeps the same direction on both sides; it does not turn.
How do you show a cubic is increasing for all $x$ ?: Show its derivative (a quadratic) is always positive, e.g. its discriminant is negative.
For which $k$ is $3x^{2}-6x+k>0$ for all $x$ ?: $k>3$ (discriminant $36-12k<0$ ).
Step 1 for finding where $f$ increases?: Differentiate, then solve $f'(x)=0$ to find the critical values.
$V'(5)=-12$ thousand m $^3$ per month. Interpret.: At $t=5$ the volume is decreasing at 12 000 m $^3$ per month.
Is it enough to say $f(3)<f(4)$ to prove $f$ is increasing?: No, you need $f'(x)>0$ on the whole interval; two values alone do not show it.