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Problem 53

$53-56$
(a) Use a graph to estimate the absolute maximum and
minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum
values.
$$f(x)=x^{5}-x^{3}+2, \quad-1 \leqslant x \leqslant 1$$

A. SEE GRAPH
The absolute maximum value of $f(x)$ is $\approx 2.19$
The absolute minimum value of $f(x)$ is $\approx 1.81$
B. Absolute minimum value is $2-\frac{6}{25} \sqrt{\frac{3}{5}} \approx 1.8141$ which occurs at $x=\frac{1}{3} \ln 2$
Absolute maximum value is $2+\frac{6}{25} \sqrt{\frac{3}{5}} \approx 2.1859$ which occurs at $x=1$

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