If the function f(x)=2x3−9ax2+12a2x+1, where a>0, attains its local maximum and local minimum at p and q respectively, such that p2=q, then f(3) is equal to:
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Step-by-Step Solution
Step 1: Find the derivative of the function
To find the local maximum and minimum points of a function, we first need to find its derivative. The derivative f'(x) gives us the slope of the tangent to the curve at any point x. We apply the power rule for differentiation to each term.
Step 2: Find critical points by setting the derivative to zero
Local maximum and minimum values occur at critical points where the derivative of the function is zero. We set f′(x)=0 and simplify the quadratic equation by dividing by 6.
Step 3: Solve for x to find p and q
We solve the quadratic equation x2−3ax+2a2=0 by factoring. The roots of this equation are the critical points where the local maximum and minimum occur. These roots are x=a and x=2a.
Step 4: Assign p and q based on the second derivative test
To determine which critical point corresponds to the local maximum (p) and which to the local minimum (q), we use the second derivative test. We find the second derivative f''(x). If f''(x) < 0, it's a local maximum; if f''(x) > 0, it's a local minimum. Since a>0, f′′(a)=−6a<0, so p=a. And f′′(2a)=6a>0, so q=2a.
Step 5: Use the condition p2=q to find the value of a
We are given the condition p2=q. Substituting the values of p=a and q=2a into this equation, we get a2=2a. Solving for a, we find two possible values: a=0 or a=2.
Step 6: Select the correct value of a and calculate f(3)
The problem states that a>0, so we discard a=0 and choose a=2. Now we substitute a=2 back into the original function f(x) to get the specific function. Finally, we evaluate f(3) by substituting x=3 into the function.