1) "f(x)=3x(x-1)^{5}"
"f'(x)=3(x-1)^5+15x(x-1)^4"
"f''(x)=15(x-1)^4+15(x-1)^4+60x(x-1)^3" "=30(x-1)^4+60x(x-1)^3"
"f'''(x)=120(x-1)^3+60(x-1)^3+180x(x-1)^2"
"=60(x-1)^2[2(x-1)+x-1+3x]"
"=60(x-1)^2[6x-3]=180(2x-1)(x-1)^2"
Hence option "C" is correct.
2) "f(x)=x^{4}-2x^{2}"
This is a polynomial function which is differentiable as well as continuous over "R"
so,
a) "f(x)" is differentiable over the interval "(-2, 2)"
b) "f(x)" is continuous over the interval "(-2, 2)"
c) "f(a)=f(2)=2^4-2\\times 2^2=8"
"f(b)=f(-2)=(-2)^4-2\\times (-2)^2=8"
"f(a)=f(b)"
So,Rolle's theorem holds good.
hence, function "f" is continuous on the closed interval "[a, b]" and differentiable on the open interval (a, b) such that"f(a) = f(b)" , then"f'(x) = 0" for some c with "a \u2264 c \u2264 b."
"f'(x)=4x^3-4x=0"
"4x(x^2-1)=4x(x-1)(x+1)=0"
"x=(-1,0,1)"
Hence option "B" is correct.
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