Question

Let f ( x ) be a polynomial of degree 6 in x , in which the coefficient of x⁶ is unity and it has extrema at x = -1 and x = 1.lim x → 0 f(x)/x³=1 then 5.f(2) is equal to

Answer 1

To find the value of 5*f(2), we need to find the value of f(2) first. Given that f(x) is a polynomial of degree 6 in x, with the coefficient of x⁶ as unity, we can express f(x) as f(x) = x⁶ + ax⁵ + bx⁴ + cx³ + dx² + ex + f. By taking the derivative of f(x) and using the fact that f'(-1) = f'(1) = 0, we can find the values of a, b, c, d, and e and then calculate f(2) and 5*f(2).

Step-by-step explanation:

To find the value of 5*f(2), we need to find the value of f(2) first. Given that f(x) is a polynomial of degree 6 in x, with the coefficient of x⁶ as unity, we can express f(x) as:

f(x) = x⁶ + ax⁵ + bx⁴ + cx³ + dx² + ex + f

Since f(x) has extrema at x = -1 and x = 1, it means that f'(-1) = f'(1) = 0. Taking the derivative of f(x), we get:

f'(x) = 6x⁵ + 5ax⁴ + 4bx³ + 3cx² + 2dx + e

Using the fact that f'(-1) = 0, we can substitute x = -1 into f'(x) and solve for a:

0 = 6(-1)⁵ + 5a(-1)⁴ + 4b(-1)³ + 3c(-1)² + 2d(-1) + e

Simplifying this equation, we get -6 + 5a - 4b + 3c - 2d + e = 0

Similarly, substituting x = 1 into f'(x), we get -6 + 5a + 4b + 3c + 2d + e = 0

Solving these two equations simultaneously, we can find the values of a, b, c, d, and e. Once we have those values, we can calculate f(2) and then find 5*f(2).