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Show that the function
f(x)=x4+8x+3 has exactly one zero in the interval [−1, 0].
Which theorem can be used to determine whether a function f(x) has any zeros in a given interval?
Determine whether there can be more than one zero in the given interval.
Rolle's Theorem states that for a function f(x) that is continuous at every point over the closed interval [a,b] and differentiable at every point of its interior (a,b), if f(a)=f(b), then there is at least one number c and (a,b) at which f′(c)=0.
Find the derivative of
f(x)=x4+8x+3.
f′(x)=
Can the derivative of f(x) be zero in the interval left bracket negative 1 comma 0 right bracket[−1, 0]?
The function f(x)=x^4+8x+3 has at least one zero at some point x=a in the interval [−1, 0]. According to Rolle's Theorem, can there be another point x bx=b in this interval where f(a)=f(b)=0?
Thus, since the intermediate value theorem shows f(x)=x4+8x+3 has at least one zero in the interval [−1, 0] and Rolle's Theorem shows that there cannot be two points x equals ax=a and x equals bx=b for which f(a)=f(b) in this interval, the function f(x) has exactly one zero in the interval [−1, 0].
