Referencing the intermediate value theorem, for a continuous f, on [a, b], f tak

Question

Referencing the intermediate value theorem, for a continuous f, on [a, b], f takes all values between f(a) and f(b). When f(a) and f(b) have opposite signs, it therefore shows that 0 lies between f(a) and f(b). Hence there are at least one zero.

If they have the same sign, it does not mean they cannot have a zero between a and b. Example: f(x) = (x-1)(x-2), and a = -10, b=10

f(-10)>0, f(10)>0, but there are two zeroes between -10 and 10

Correct there are two zeros. Do you know what they are?

Explanation / Answer

f(x) = (x-1)(x-2)

=> f(-10)= 132

=> f(10)= 72

As f(10)< f(-10), there has to be a zero in between x= -10 and x=10

To find the zero, we use algebra and equate the function to 0.

=> (x-1)(x-2) = 0

=> x=1, x=2

=> The two zeros in between x= -10 and x=10 are x- 1, x=2.... Answer

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