Porisma

Worked solution

Differentiate turning points and a stationary inflection

The diagram shows y=f(x)y=f(x) with a maximum at x=ax=a, minimum at x=bx=b, and stationary point of inflection at the origin. On a separate diagram draw y=f′(x)y=f^{\prime}(x). marks

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Given function rises to a maximum at negative x=a, decreases through a stationary inflection at the origin, continues to a minimum at positive x=b, then rises. The two turning-point x-values are marked a and b without numerical coordinates.
Given function rises to a maximum at negative x=a, decreases through a stationary inflection at the origin, continues to a minimum at positive x=b, then rises. The two turning-point x-values are marked a and b without numerical coordinates.
Given function rises to a maximum at negative x=a, decreases through a stationary inflection at the origin, continues to a minimum at positive x=b, then rises. The two turning-point x-values are marked a and b without numerical coordinates.

[2009/SRJC/Promo/2]

CJC · Review problems · Question 3

Answer

Zeros at a,0,ba,0,b: positive-to-negative crossing at aa, touch from below at 00, negative-to-positive crossing at bb.

Place the three zeros and their sign changes

At a<0a<0, f′f^{\prime} crosses from positive to negative. At b>0b>0, it crosses from negative to positive. At the origin, the stationary inflection makes f′(0)=0f^{\prime}(0)=0 but the function keeps decreasing, so f′f^{\prime} touches the axis from below rather than crossing.

How did this part go?

Retain the intervening source branches

The printed derivative has an unlabelled positive maximum before aa, a negative minimum between aa and 00, a local maximum at (0,0)(0,0), and another negative minimum between 00 and bb. It then rises through (b,0)(b,0). These qualitative extrema follow the slope changes drawn in the source; their numerical coordinates are not given.

How did this part go?

Compare your sketch

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Derivative has zeros a,0,b. It is positive before a, negative between a and b except for its touch at the origin, then positive after b. The source shape includes a positive peak before a, negative troughs between a and zero and between zero and b, and a local maximum at the origin.
Derivative has zeros a,0,b. It is positive before a, negative between a and b except for its touch at the origin, then positive after b. The source shape includes a positive peak before a, negative troughs between a and zero and between zero and b, and a local maximum at the origin.
Derivative has zeros a,0,b. It is positive before a, negative between a and b except for its touch at the origin, then positive after b. The source shape includes a positive peak before a, negative troughs between a and zero and between zero and b, and a local maximum at the origin.

Common mistakes

Do not force the derivative to cross the axis at every stationary point.

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