In order to find the points of extremum of the given integral, we start by considering the integral:
x2dtLet's denote this integral by
F(x).
We need to find the critical points of
F(x), which are the points where its derivative with respect to
x is zero. Since
F(x) is an integral with an upper limit that depends on
x, we will use the Leibniz rule for differentiation under the integral sign:
In our case, the lower limit is 0 and is independent of
x, so its derivative is zero. The function inside the integral does not explicitly depend on
x, so the partial derivative with respect to
x is zero. Hence, we simplify to:
Since the value of the integral at the lower limit is zero, we get:
F(x)=⋅2xSimplifying the expression inside the fraction, we have:
F(x)=⋅2x To find the critical points, we must set the derivative equal to zero:
⋅2x=0The fraction is zero if and only if the numerator is zero. Thus, we solve:
(x4−5x2+4)⋅2x=0This simplifies to two cases:
2x(x4−5x2+4)=0From the above equation, set each factor to zero:
2x=0⇒x=0and
x4−5x2+4=0Letting
y=x2, we convert the quartic equation to a quadratic one:
y2−5y+4=0Solving this quadratic equation by factoring:
(y−1)(y−4)=0So,
y=1 or y=4Since
y=x2, we have:
x2=1⇒x=±1x2=4⇒x=±2 Thus, the points of extremum are:
Option A:
±1Option B:
±2Therefore, the correct answer is both Option A and Option B.