Analyzing the Derivative of a Given Function | Finding the True Statement

Let f be the function given by f(x)=1/7×7+1/2×6−x5−15/4×4+4/3×3+6×2. Which of the following statements is true?

To determine which of the following statements is true, let’s first simplify the given function f(x):

f(x) = (1/7)x^7 + (1/2)x^6 – x^5 – (15/4)x^4 + (4/3)x^3 + 6x^2

Now we can analyze the given options and test them against the function:

a) f'(x) = 0 for all x
b) f'(x) = 0 only for x = 0
c) f'(x) = 0 only for x = 1
d) f'(x) = 0 for x = 0 and x = 1

To find the derivative of f(x), we will use the power rule and properties of differentiation:

f'(x) = (7/7)x^(7-1) + (6/2)x^(6-1) – (5/1)x^(5-1) – (4/4)x^(4-1) + (3/3)x^(3-1) + 2(6)x^(2-1)
= x^6 + 3x^5 – 5x^4 + 3x^2 + 12x

Now let’s examine each option:

a) f'(x) = 0 for all x

If this statement is true, then the derivative f'(x) would have to equal zero for every x value

To determine which of the following statements is true, let’s first simplify the given function f(x):

f(x) = (1/7)x^7 + (1/2)x^6 – x^5 – (15/4)x^4 + (4/3)x^3 + 6x^2

Now we can analyze the given options and test them against the function:

a) f'(x) = 0 for all x
b) f'(x) = 0 only for x = 0
c) f'(x) = 0 only for x = 1
d) f'(x) = 0 for x = 0 and x = 1

To find the derivative of f(x), we will use the power rule and properties of differentiation:

f'(x) = (7/7)x^(7-1) + (6/2)x^(6-1) – (5/1)x^(5-1) – (4/4)x^(4-1) + (3/3)x^(3-1) + 2(6)x^(2-1)
= x^6 + 3x^5 – 5x^4 + 3x^2 + 12x

Now let’s examine each option:

a) f'(x) = 0 for all x

If this statement is true, then the derivative f'(x) would have to equal zero for every x value. However, looking at f'(x) = x^6 + 3x^5 – 5x^4 + 3x^2 + 12x, this is not the case. Therefore, option a) is not true.

b) f'(x) = 0 only for x = 0

To verify this statement, we need to check if f'(x) = 0 only when x = 0. From f'(x) = x^6 + 3x^5 – 5x^4 + 3x^2 + 12x, it is evident that f'(x) can take the value zero for x = 0. However, it can also equal zero for other values of x, such as x = -2 or x = 1. Therefore, option b) is not true.

c) f'(x) = 0 only for x = 1

Similarly, we can evaluate if f'(x) = 0 only when x = 1. However, using the same derivative f'(x) = x^6 + 3x^5 – 5x^4 + 3x^2 + 12x, we can see that f'(x) can also equal zero for other values of x, such as x = -3 or x = 2. Therefore, option c) is not true.

d) f'(x) = 0 for x = 0 and x = 1

Now let’s examine if f'(x) = 0 for both x = 0 and x = 1. Evaluating f'(x) = x^6 + 3x^5 – 5x^4 + 3x^2 + 12x at x = 0 and x = 1, we find:

f'(0) = 0^6 + 3(0)^5 – 5(0)^4 + 3(0)^2 + 12(0) = 0
f'(1) = 1^6 + 3(1)^5 – 5(1)^4 + 3(1)^2 + 12(1) = 10

So, f'(x) = 0 for x = 0, but f'(x) ≠ 0 for x = 1. Therefore, option d) is also not true.

To summarize, none of the given statements a), b), c), or d) are true for the given function f(x). The correct option might not be included among the options provided.

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