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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 1

\b[No.] On $[-1,1]$, take
$$
f(x)=x^3,\qquad \xi=0.
$$
Then $f'(0)=0$. For every $x_1<0<x_2$,
$$
\frac{f(x_2)-f(x_1)}{x_2-x_1}
=x_2^2+x_1x_2+x_1^2.
$$
This quadratic expression is strictly positive: it equals
$$
\left(x_2+\frac{x_1}{2}\right)^2+\frac34x_1^2.
$$
\b[Thus no chord with endpoints on opposite sides of $\xi$ has slope $f'(\xi)=0$.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 11D / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 11D / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 11D / c / Solution

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