Part of the graph of the function f(x) = (x + 4)(x – 6) is shown below. Which statements about the function are true? Select two options.
Answers
Given : graph of the function f(x) = (x + 4)(x – 6)
To Find : Which statements about the function are true
The vertex of the function is at (1,–25).
The vertex of the function is at (1,–24).
The graph is increasing only on the interval −4< x < 6.
The graph is positive only on one interval, where x < –4.
The graph is negative on the entire interval
–4 < x < 6.
Solution:
f(x) = (x + 4)(x – 6)
f'(x) = (x + 4) + (x - 6) = 2x - 2
2x - 2 = 0
=> x = 1
f''(x) = 2 > 0 Hence minimum value at x = 1
f(1) = (1 + 4)(1 - 6) = - 25
Hence vertex = ( 1 , - 25)
The vertex of the function is at (1,–25). is CORRECT
f(x) = (x + 4)(x – 6)
x > 6 and x < - 4 f(x) > 0 The graph is positive only on one interval, where x < –4. is INCORRECT as its for x > 6 also
x = 6 and x = - 4 f(x) = 0
-4 < x < 6 f(x) = - ve
The graph is negative on the entire interval -4 < x < 6. is CORRECT
f'(x) = 2x - 2 ≥ 0 if x ≥ 1
Hence graph is increasing for x ≥ 1
The graph is increasing only on the interval −4< x < 6. is INCORRECT
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