Question - Analyzing Statements with a Decreasing Function Graph
Solution:
The image shows a graph of a function f(x) which is decreasing on the interval [a, b]. The question asks which of the statements provided is incorrect given the graph.A) x · f'(x) < 0: Since x is positive on the interval [a, b] and f'(x) is negative (because the function is decreasing), their product is indeed negative. Therefore, this statement is correct.B) f(x) + f'(x) < 0: Since f(x) is positive (above the x-axis) and f'(x) is negative (the slope of the graph is negative), their sum might be negative or positive depending on the magnitudes. We can't definitively determine the sign of the sum without more information. This one cannot immediately be identified as false without further analysis, so we'll move on and come back if necessary.C) x - f'(x) > 0: We know x is positive and f'(x) is negative. Subtracting a negative number is equivalent to adding a positive number, so x - f'(x) would indeed be greater than 0. Therefore, this statement is correct.D) f(x) · -f'(x) > 0: Since f(x) is positive and -f'(x) is positive (because f'(x) is negative, and the negative of a negative is positive), their product will be positive. Therefore, this statement is correct.E) [f'(x)]^2 < 0: The square of any real number is non-negative. Since f'(x) is real, squaring it will give a non-negative result, so this statement is false.The incorrect statement is E) [f'(x)]^2 < 0.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com