Find the maximum and minimum values, if any of the function f(x) = -(x – 1)² + 10.

f(x) = -(x – 1)² + 10

Maximum or minimum values of a function occur at values of x where f’(x) = 0

Let us find the points where f’(x) = 0

Differentiate f(x) and equate to 0

⇒ f’(x) = -2(x – 1)

⇒ 0 = -2x + 2

⇒ 2x = 2

⇒ x = 1

Now to check whether x = 1 is a point of minima or maxima we have to check behaviour of f’’(x). If f’’(x) is negative at x = 1(that is f’’(1) < 0) then x = 1 is a point of maxima and else if f’’(x) is positive at x = 1(that is f’’(1) > 0) then x = 1 is a point of minima

We have calculated earlier f’(x) as

⇒ f’(x) = -2(x – 1)

⇒ f’(x) = -2x + 2

Differentiate again

⇒ f’’(x) = -2

f’’(x) is a constant function

⇒ f’’(1) = -2

As f’’(1) is negative hence x = 1 is a point of maxima

Hence the maximum value of f(x) will be at x = 1

Put x = 1 in f(x)

⇒ f(1) = -(1 – 1)² + 10

⇒ f(1) = 10

Hence 10 is the maximum value of given function and there is no minimum value