Is #f(x)=-4x^2+x-1# increasing or decreasing at #x=1#?

You must ascertain the value of f'(a) in order to ascertain whether a function is increasing or decreasing at x = a.

• At x = a, f(x) is increasing if f'(a) > 0.

• At x = a, f(x) is decreasing if f'(a) < 0.

f'(1) = -8 + 1 = -7, and

since f'(1) < 0 , then f(x) is decreasing at x = 1 graph{-4x^2+x-1 [-10, 10, -5, 5]}

To determine whether the function ( f(x) = -4x^2 + x - 1 ) is increasing or decreasing at ( x = 1 ), we need to examine the sign of the derivative of the function at that point. The derivative of ( f(x) ) with respect to ( x ) is given by ( f'(x) = -8x + 1 ).

Evaluate ( f'(1) ). If ( f'(1) > 0 ), the function is increasing at ( x = 1 ). If ( f'(1) < 0 ), the function is decreasing at ( x = 1 ).

Substitute ( x = 1 ) into ( f'(x) ): ( f'(1) = -8(1) + 1 = -8 + 1 = -7 ).

Since ( f'(1) = -7 < 0 ), the function ( f(x) = -4x^2 + x - 1 ) is decreasing at ( x = 1 ).