3. (5pts) find any maxima or minima (function values and points x where they occur) of the following…

3. (5pts) find any maxima or minima (function values and points x where they occur) of the following function $-2x^{2}+4x + 3$

3. (5pts) find any maxima or minima (function values and points x where they occur) of the following function $-2x^{2}+4x + 3$

Answer

Explanation:

Step1: Find the first derivative

The function is ( f(x) = -2x^2 + 4x + 3 ). The first derivative ( f'(x) ) is found using the power rule. For ( ax^n ), the derivative is ( nax^{n - 1} ). So, ( f'(x)=\frac{d}{dx}(-2x^2 + 4x + 3)=-4x + 4 ).

Step2: Find critical points

Set ( f'(x) = 0 ) to find critical points: ( -4x + 4 = 0 ) Subtract 4 from both sides: ( -4x=-4 ) Divide both sides by -4: ( x = 1 )

Step3: Find the second derivative

The second derivative ( f''(x) ) of ( f(x) = -2x^2 + 4x + 3 ) is ( f''(x)=\frac{d}{dx}(-4x + 4)=-4 )

Step4: Determine if it's a maxima or minima

Since ( f''(x)=-4<0 ), the function has a local maximum at ( x = 1 ).

Step5: Find the function value at ( x = 1 )

Substitute ( x = 1 ) into ( f(x) ): ( f(1)=-2(1)^2 + 4(1)+3=-2 + 4 + 3 = 5 )

Answer:

The function ( f(x)=-2x^2 + 4x + 3 ) has a local maximum of ( 5 ) at ( x = 1 ). There is no local minimum.