Answer to Question #247698 in Real Analysis for Sree

Question

Verify Maclaurin's theorem for f(x)=(1-x)^(5/2) with Lagrange's form of remainder upto 3 terms where x=1

Answer 1

$f(x)=f(0)+f'(0)x+{\frac {f''(0)x^{2}} 2} + {\frac {f'''(c)x^{3}} 6}$

where $c\in (0;x)$, $x=1$

f(1) = 0

f(0) = 1

$f'(x) = -{\frac 5 2} (1-x)^{{\frac 3 2}}$

f'(0) = -2.5

$f''(x) = {\frac {15} 4} (1-x)^{{\frac 1 2}}$

f''(0) = 3.75

f'''(c) = $-{\frac {15} {8\sqrt{1-c}}}$

So, we have to verify if $\exist c\in (0;1): 0=1-2.5+1.875-{\frac {15} {48\sqrt{1-c}}}$

${\frac {15} {48\sqrt{1-c}}}=0.375$

$\sqrt{1-c} = {\frac 5 6}$

$c = {\frac {11} {36}}$

Theorem has been verified and proved.