The average rate of change of f(x) = a $x^{2}$ +bx+c , over the interval [x1, x2] is zero. Then x1 and x2 are the roots of f(x) = 0

True

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False

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Solution

The correct option is B
False

We saw that average rate of change is the ratio of change in f(x) to change in x. That is $\frac{f (a + h) - f (a)}{h}$ over the interval [a, a+h]. We are given that the average rate of change over the interval [x1,x2] is zero. That means the numerator in the expression f(a+h) - f(a) is zero. In our case, this will be f(x2) - f(x1).

=> f(x2) - f(x1) = 0

=> f(x2) = f(x1)

The given statement is, x1 and x2 are the roots of the given quadratic. We can see that this is not necessarily/always true. If they are roots, then f(x1) = f(x2) = 0. In this case also, the average rate of change will be zero. But it is not necessary that x1 and x2 are the roots for the average rate to be zero.

For example if take the quadratic y = $x^{2}$ the average rate of change over the intervals, [-1,1], [-2,2], [-2.2, 2.2] will be zero, even if they are not the roots of given equation

So, the given statement is false

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The average rate of change of f(x) = a $x^{2}$ +bx+c , over the interval [x1, x2] is zero. Then x1 and x2 are the roots of f(x) = 0