Question

Can you guys walk me through how to find the local maximum(s) of the following function $f(x) = \int_{-20}^{x}(t-5)^3(t+3)^2(t-2)^6(t+7)^9(t-\pi)dt$

Answer 1

fe) - fasicos 4-51°C++3)*(4-2)*(+27) P4-6) 4t -20 f derung fundamental theoream of Calculas. df(x) da 2-513 (243)*(x-2) (377)9 (-) Pest df(x) -D da 62-5)3 (243)(x-2)6 (2+7)º(x-ā) = 0 So solutions are x=5,-3, 2, 7, Ã d²f(x) dx² (x+5)*(7+3)*(x-3)* {x17) + 3(7-5)+(x+372 (2-2)*_*++)'(2-a) + 2(7-1/" [*13) (2-2) &173(2-2) + ((**)*(+5)*c*-220112) to 96-513 (2+3)2(1-2) 6 (347) '(2-5/

Now, df(x) daz + (4-5)*(243)*(x-2)6(847) 9 3(2-s)2(x+3) (7-2)6(+7)*(x-1) + 2(+5)° (x+3) (x-2)6 (247)9(-ā) + 6(7-5)°(2+3)*(x-2)5(X+7)964-) 9 (2-5)° (x+3)(x-2)*(*+7)® (-a) Now at 2=15 a²f(a) daz =0 O at x=-3 at x=2 o dx² d²fcas da? dita) da? at a=-7

at a= all other terms will be zero at x=ã except first d4(2), (85)3 CT +3) ? (7-2)' (+ 7)9 da² Positive Positive Positive negative So, 04(x) at x=T da² Local so, & maaimia at x =ī