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Ta có:\(f\left(x\right)=x^8-100x^7-x^7+100x^6-....+x^2-100x-x+100-75\)
\(=x^7\left(x-100\right)-x^6\left(x-100\right)-....+x\left(x-100\right)-\left(x-100\right)-75\)
Nên \(f\left(100\right)=x^7.\left(100-100\right)-x^6\left(100-100\right)-....+x\left(100-100\right)-\left(100-100\right)-75\)
\(=-75\)
Với x= 100 thì 101=x+1 nên ta có f(100)=x\(^8\)-(x+1)x\(^7\)=(x+1)x\(^6\)-(x+1)x\(^5\)+....-(x+1)+25=x\(^8\)-x\(^8\)+x\(^7\)-......-x-1+25=24
\(\text{1)}\)
\(\text{Thay }x=-2,\text{ ta có: }f\left(-2\right)-5f\left(-2\right)=\left(-2\right)^2\Rightarrow f\left(-2\right)=-1\)
\(\Rightarrow f\left(x\right)=x^2+5f\left(-2\right)=x^2-5\)
\(f\left(3\right)=3^2-5\)
\(\text{2)}\)
\(\text{Thay }x=1,\text{ ta có: }f\left(1\right)+f\left(1\right)+f\left(1\right)=6\Rightarrow f\left(1\right)=2\)
\(\text{Thay }x=-1,\text{ ta có: }f\left(-1\right)+f\left(-1\right)+2=6\Rightarrow f\left(-1\right)=2\)
\(\text{3)}\)
\(\text{Thay }x=2,\text{ ta có: }f\left(2\right)+3f\left(\frac{1}{2}\right)=2^2\text{ (1)}\)
\(\text{Thay }x=\frac{1}{2},\text{ ta có: }f\left(\frac{1}{2}\right)+3f\left(2\right)=\left(\frac{1}{2}\right)^2\text{ (2)}\)
\(\text{(1) - 3}\times\text{(2) }\Rightarrow f\left(2\right)+3f\left(\frac{1}{2}\right)-3f\left(\frac{1}{2}\right)-9f\left(2\right)=4-\frac{1}{4}\)
\(\Rightarrow-8f\left(2\right)=\frac{15}{4}\Rightarrow f\left(2\right)=-\frac{15}{32}\)
Tính \(f\left(1\right)\)
\(f\left(x\right)=1+x^3+x^5+x^7+...+x^{101}\)
\(\Rightarrow f\left(1\right)=1+1^3+1^5+1^7+...+1^{101}\)
\(=1+1+1+1+...+1\) (có \(51\) số \(1\))
\(=51\)
Tính \(f\left(-1\right)\)
\(f\left(x\right)=1+x^3+x^5+x^7+...+x^{101}\)
\(\Rightarrow f\left(-1\right)=1+\left(-1\right)^3+\left(-1\right)^5+...+\left(-1\right)^{101}\)
\(=1+\left(-1\right)+\left(-1\right)+\left(-1\right)+...+\left(-1\right)\) (có \(50\) số \(-1\))
\(=1+\left(-50\right)\)
\(=-49\)
Vậy: \(\left\{{}\begin{matrix}f\left(1\right)=51\\f\left(-1\right)=-49\end{matrix}\right.\)
Ta có:
a) \(f\left(1\right)=1+1^3+1^5+1^7+...+1^{101}\)
\(f\left(1\right)=1+50=51\)
b) \(f\left(-1\right)=1+\left(-1\right)^3+\left(-1\right)^5+\left(-1\right)^7+...+\left(-1\right)^{101}\)
\(f\left(-1\right)=1-50=-49\)