Show that the function f(x) = x
3 +
4
x2 + 7 has exactly one zero in the interval
(−∞, 0)
Answers
Answer:
f(x) has exactly one zero in the interval (−∞, 0).
Step-by-step explanation:
f(x) = x³ + 4x² + 7
For roots f(x) =0
x³ + 4x² + 7 = 0 -----------(1)
now differentiating (1) we get
3x² + 8x = 0
=> x = 0, -8/3
=> The function f(x) is increasing in (-∞, -8/3) ∪ (0, ∞) and decreasing in (-8/3, 0).
f(0) = 7 which is positive number.(local minima)
Again moving in positive X-direction f(x) will increase as its derivative is positive in (0, ∞). So, f(x) will not cut the X-axis in (0, ∞).
Now,
f(-8/3) = 16.48 which is positive. (local maxima)
f(-∞) = -∞ which is negative. This means the curve must have crossed the x-axis somewhere between -∞ and 0.
(If any confusion take help of the attached image)
Hence, f(x) has exactly one zero in the interval (−∞, 0).
