Consider the function

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

F(x)=x^{6}-3a x^{4}+b x^{2}

, where

a

and

b

are real constants. (i) Show that

x=0

is a stationary point of

F

for all real values of

a

and

b

. (ii) Find

F''(0)

and hence state the condition on

b

for which the second derivative test is inconclusive at

x=0

. (iii) When the condition from part (ii) holds, examine the next non-zero derivative of

F

x=0

and determine, in terms of

a

, whether the stationary point is a local maximum, a local minimum, or neither. (iv) For those values of

a

and

b

that make

x=0

a local minimum, compute the minimum value of

F

Related theory

Curve Analysis
Higher-Order Derivatives
Advanced Differentiation
Second order derivatives
Continuity and Differentiability

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