A branch of an implicitly defined curve passes through the point

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

(2,5)

. Along this branch, the derivative

\dfrac{dy}{dx}

satisfies the following sign chart with respect to the

x

–coordinate: • For

x<2

\dfrac{dy}{dx}<0

. • At

x=2

\dfrac{dy}{dx}=0

. • For

x>2

\dfrac{dy}{dx}>0

. (1) Without performing any explicit differentiation, explain what the sign information given above reveals about the monotonic behavior of

y

x

moves through the value

2

along the branch. (2) Using your explanation from part (a), determine whether the point

(2,5)

represents a relative minimum, a relative maximum, or neither for the

y

–coordinate with respect to

x

. Justify your answer thoroughly in words. (3) Describe how the local geometric shape of the curve near

(2,5)

is reflected by the sign change of

\dfrac{dy}{dx}

. In your description, refer to the slope of the tangent line and the orientation of the curve before and after

x=2

. (A sketch is not required but may be included if it aids your explanation.)