TMUA 2016 Paper 2 Q11 — Worked Solution

P2·Differentiation, Functions, Transformations & Graphs +1·6.5Official · 2016 · Q11

polynomial-rootsrolles-theoremreflectionmust-be-true

f (x)

is a polynomial with real coefficients. The equation

f (x) = 0

has exactly two real roots,

x = - p

and

x = p

, where

p > 0

. Consider the following three statements:

f^{'} (x) = 0

for exactly one value of

x

between

- p

and

p

The area between the curve

y = f (x)

, the

x

-axis and the lines

x = - p

and

x = p

is given by

2 \int_{0}^{p} f (x) d x

III

The graph of

y = - f (- x)

intersects the

x

-axis at the points

x = - p

and

x = p

only

Which of the above statements must be true?

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