If f(x)=1.7x^5-5.3x^2+9.8 and if r is the only real number such that f(r)=0, then r is between
a. -1.4 and -1.3
b. -1.3 and -1.2
c. -1.2 and -1.1
d. -1.1 and 0
e. -1.0 and -0.9
To find the value of r, we need to solve the equation f(r) = 0. In this case, our function f(x) is given as f(x)=1.7x^5-5.3x^2+9.8.
To solve f(r) = 0, we can set the equation equal to zero and solve for r:
1.7r^5 - 5.3r^2 + 9.8 = 0
Since this is a polynomial equation, we may not be able to find an exact solution for r. However, we can use numerical methods or approximation techniques to estimate the value of r. One popular numerical method is the Newton-Raphson method.
But to estimate the range of r, we can use the Intermediate Value Theorem. According to the theorem, if a polynomial is continuous on an interval [a, b] and takes on positive and negative values at the endpoints, there must be at least one real root in that interval.
Let's evaluate f(-1.4) and f(-1.3) to see if they have opposite signs:
f(-1.4) = 1.7(-1.4)^5 - 5.3(-1.4)^2 + 9.8 ≈ -37.3692
f(-1.3) = 1.7(-1.3)^5 - 5.3(-1.3)^2 + 9.8 ≈ 15.6061
Since f(-1.4) is negative and f(-1.3) is positive, they have opposite signs. Therefore, based on the Intermediate Value Theorem, we can conclude that there is a real root of f(x) = 0 between -1.4 and -1.3. Hence, the answer is option (a) -1.4 and -1.3.