IIT-JEE 2011 · Paper 1 · Mathematics Q66

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MathematicsInteger+4Question 66 of 69

Let $f:[1,\infty)\to[2,\infty)$ be a differentiable function such that $f(1)=2$ . If
$6\int_1^xf(t)\,dt=3xf(x)-x^3$
for all $x\ge1$ , then the value of $f(2)$ is

Your answer

Answer and worked solution

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The data are inconsistent. At $x=1$ the equation gives $0=3f(1)-1$ , i.e. $f(1)=\frac13$ , contradicting $f(1)=2$ .

Formal working. Differentiating, $6f=3f+3xf'-3x^2$ , so $f'-\frac fx=x$ and $f=x^2+Cx$ . Neither $f(1)=2$ ( $C=1$ , $f(2)=6$ ) nor the integral equation itself ( $C=-\frac23$ , which also violates $f\ge2$ ) gives a consistent answer.

Topics:Linear first-order equations Fundamental theorem of calculus

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IIT-JEE 2011 · Paper 1, Mathematics question 66