Let f:R→R be a polynomial function of degree four having extreme values at x=4 and x=5. If x→0limx2f(x)=5, then f(2) is equal to:
Get the complete, step-by-step math solution for: "Let f: {R} {R} be a polynomial function of degree four having extreme values at x = 4 and x = 5. If _{x 0} (f(x))/(x²) = 5, then f(2) is equal to:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define the polynomial and its derivative
We are given that f(x) is a polynomial of degree four. We can write its general form as f(x)=ax4+bx3+cx2+dx+e. The extreme values occur where the first derivative f'(x) is zero. Let's find the derivative of f(x).
Step 2: Use the limit condition to find coefficients
We are given the limit limx→0x2f(x)=5. For this limit to be finite and non-zero, the terms dx and e in the numerator must be zero, otherwise the limit would be infinite or undefined. After setting d=0 and e=0, we can divide by x2 and take the limit, which gives us c=5.
Step 3: Formulate the derivative based on extreme values
Since f(x) has extreme values at x=4 and x=5, these are roots of f′(x)=0. Also, from the previous step, we know d=0, so f′(0)=0. This means x=0 is also a root of f′(x)=0. Therefore, f'(x) must have factors (x−4), (x−5), and x. We can write f′(x)=kx(x−4)(x−5) for some constant k. Expanding this, we get f′(x)=k(x3−9x2+20x).
Step 4: Equate coefficients to find 'a' and 'b'
By comparing the coefficients of the two expressions for f'(x), 4ax3+3bx2+10x and k(x3−9x2+20x), we can solve for k, a, and b. From the coefficient of x, we find k=1/2. Then, substituting k back, we find a=1/8 and b=−3/2.
Step 5: Write the complete polynomial function
Now that we have found the values of a, b, c, d, and e, we can write the complete polynomial function f(x). Substituting a=1/8, b=−3/2, c=5, d=0, and e=0 into the general form, we get f(x)=81x4−23x3+5x2.
Step 6: Calculate f(2)
Finally, we need to find the value of f(2). Substitute x=2 into the polynomial function we just found and simplify the expression.