Diketahui g(x)=31​x3−A2x+1; f(x) = g(2x - 1), A suatu konstanta. Jika f naik pada x≤0 atau x≥1, nilai maksimum relatif g adalah...

Pertanyaan

Diketahui begin mathsize 12px style g left parenthesis x right parenthesis equals 1 third x cubed minus A squared x plus 1 end style; f(x) = g(2x - 1), A suatu konstanta. Jika f naik pada xbegin mathsize 12px style less or equal than end style0 atau xbegin mathsize 12px style greater or equal than end style1, nilai maksimum relatif g adalah... 

  1. begin mathsize 12px style 7 over 3 end style

  2. begin mathsize 12px style 5 over 3 end style

  3. begin mathsize 12px style 1 third end style

  4. begin mathsize 12px style negative 1 third end style

  5. begin mathsize 12px style negative 5 over 3 end style

Y. Laksmi

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Pembahasan

Jika begin mathsize 12px style g left parenthesis x right parenthesis equals 1 third x cubed minus A squared x plus 1 end style maka diperoleh:

begin mathsize 12px style f left parenthesis x right parenthesis equals g open parentheses 2 x minus 1 close parentheses equals 1 third open parentheses 2 x minus 1 close parentheses cubed minus A squared open parentheses 2 x minus 1 close parentheses plus 1 end style

Fungsi akan naik jika f'(x) > 0, sehingga:

begin mathsize 12px style f apostrophe left parenthesis x right parenthesis greater than 0  1 third times 3 open parentheses 2 x minus 1 close parentheses squared times 2 minus 2 A squared greater than 0  2 open parentheses 4 x squared minus 4 x plus 1 close parentheses minus 2 A squared greater than 0  8 x squared minus 8 x plus 2 minus 2 A squared greater than 0  x squared minus x plus fraction numerator 1 minus A squared over denominator 4 end fraction greater than 0 end style

Jika f naik pada xbegin mathsize 12px style less or equal than end style0 atau xbegin mathsize 12px style greater or equal than end style1, maka akar-akar persamaan yang dapat terbentuk yaitu x(x - 1)begin mathsize 12px style greater or equal than end style0.

Cari nilai A dengan membandingkan kedua persamaan:

begin mathsize 12px style open curly brackets table attributes columnalign left end attributes row cell x squared minus x plus fraction numerator 1 minus A squared over denominator 4 end fraction greater than 0 end cell row cell x left parenthesis x minus 1 right parenthesis greater or equal than 0 end cell end table close  x squared minus x plus fraction numerator 1 minus A squared over denominator 4 end fraction equals x left parenthesis x minus 1 right parenthesis  x squared minus x plus fraction numerator 1 minus A squared over denominator 4 end fraction equals x left parenthesis x minus 1 right parenthesis  x squared minus x plus fraction numerator 1 minus A squared over denominator 4 end fraction equals x squared minus x  fraction numerator 1 minus A squared over denominator 4 end fraction equals 0  1 minus A squared equals 0  A squared equals 1  end style

Sehingga diperoleh

begin mathsize 12px style g open parentheses x close parentheses equals 1 third x cubed minus A squared x plus 1  g left parenthesis x right parenthesis equals 1 third x cubed minus x plus 1 end style

 

Mencari titik stasioner g'(x) = 0;

g'(x) = 0

begin mathsize 12px style x squared minus 1 equals 0  end style

(x - 1)(x + 1) = 0

x = 1 atau x = -1

Nilai maksimum relatif jika g"(x) < 0:

g"(x) = 2x

g"(-1) = 2(-1) = -2

g(1) = 2(1) = 2

Diperoleh titik maksimum relatif saat x = -1.

Nilai maksimum relatif fungsi:

begin mathsize 12px style g left parenthesis negative 1 right parenthesis equals 1 third open parentheses negative 1 close parentheses cubed minus open parentheses negative 1 close parentheses plus 1  g left parenthesis negative 1 right parenthesis equals negative 1 third plus 2  g left parenthesis negative 1 right parenthesis equals negative 1 third plus 6 over 3  g left parenthesis negative 1 right parenthesis equals 5 over 3 end style

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