Let , where and are differentiable on an open interval and for all . Which of the…

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

r(x)=\dfrac{p(x)}{q(x)}

, where

p

and

q

are differentiable on an open interval

I

and

q(x)\neq 0

for all

x\in I

. Which of the following best explains why the equation

p'(x)q(x)-p(x)q'(x)=0

identifies all interior points of

I

at which

r

could attain a local extremum? (a) Because the equation forces

p'(x)

and

q'(x)

to have opposite signs, meaning one function is increasing while the other is decreasing, so their ratio must reach a local maximum or minimum at that point. (b) Because the product rule gives

(p\,q)'=p'(x)q(x)+p(x)q'(x)

; if

p'(x)q(x)-p(x)q'(x)=0

, then

(p\,q)'=0

, so

p(x)q(x)

is locally constant and therefore the quotient

r

has an extremum. (c) Because

r'(x)=\dfrac{p'(x)q(x)-p(x)q'(x)}{[q(x)]^{2}}

, setting the numerator equal to

0

forces

r'(x)=0

whenever the denominator is non-zero, giving the only possible horizontal tangents. (d) Because

r'(x)=\dfrac{p'(x)q(x)-p(x)q'(x)}{[q(x)]^{2}}

, the given equation makes the denominator of

r'(x)

equal to

0

, so the derivative is undefined and

r

must change direction there, producing an extremum.

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